Domination in Graphs: Core Concepts 3031094956, 9783031094958

Table of contents :
Dedication
Preface
Contents
Chapter 1 In the Beginning: Roots of Domination in Graphs
1.1 Introduction
1.2 Basic Terminology
1.3 Origins
1.3.1 Defensive and Offensive Strategies of the Roman Empire
1.3.2 Chaturanga
1.3.3 Eight Queens Problem
1.3.4 Five Queens Problem
1.3.5 Queens Independent Domination Problem
1.3.6 Queens Total Domination Problem
1.3.7 Generalizations to Other Chess Pieces
1.4 Application Driven Origins
1.4.1 Radio Broadcasting
1.4.2 Computer Communication Networks
1.4.3 Sets of Representatives
1.4.4 School Bus Routing and Bus Stop Selection
1.4.5 Electrical Power Domination
1.4.6 Influence in Social Networks
1.4.7 Topographic Maps
1.4.8 Transporting Hazardous Materials
1.5 Early Chronology of Domination in Graph Theory
Chapter 2 Fundamentals of Domination
2.1 Introduction
2.2 Core Concepts
2.2.1 Independent Sets
2.2.2 Dominating Sets
2.2.3 Irredundant Sets
2.3 Parameters Suggested by the Definition of a Dominating Set
2.3.1 Total Dominating Sets
2.3.2 k-Dominating Sets
2.3.3 H-forming Dominating Sets
2.3.4 Perfect and Efficient Dominating Sets
2.3.5 Distance-k Dominating Sets
2.3.6 Fractional Domination
2.4 Equivalent Formulations of Domination
2.4.1 Pendant Edges in Spanning Forests
2.4.2 Enclaveless Sets
2.4.3 Spanning Star Partitions
2.4.4 Non-dominating Partitions
2.4.5 Total Domination and Splitting Graphs
2.4.6 Dominating Sets and (1, 4 : 3)-Sets
2.5 Domination in Terms of Perfection in Graphs
2.6 Ore’s Lemmas and Their Implications
Chapter 3 Complexity and Algorithms for Domination in Graphs
3.1 Introduction
3.2 Brief Review of NP-Completeness
3.3 NP-Completeness of Domination, Independent Domination, and Total Domination
3.3.1 NP-Completeness Results for Arbitrary Graphs
3.3.2 NP-Completeness Results for Bipartite Graphs
3.3.3 Summary of Complexity Results for Graph Families
3.4 A Representative Sample of Domination Algorithms for Trees
3.4.1 Minimum Dominating Set
3.4.2 Minimum Independent Dominating Set
3.4.3 Minimum Total Dominating Set
3.5 Early Domination Algorithms and NP-Completeness Results
3.6 Other Sources for Domination Algorithms and Complexity
Chapter 4 General Bounds
4.1 Introduction
4.2 Domination and Maximum Degree
4.2.1 Domination Number and Maximum Degree
4.2.2 Total Domination Number and Maximum Degree
4.2.3 Independent Domination Number and Maximum Degree
4.3 Domination and Order
4.3.1 Domination Number and Order
4.3.2 Total Domination Number and Order
4.3.3 Independent Domination Number and Order
4.4 Basic Relationships Among Core Parameters
4.5 Domination and Distance
4.6 Domination and Packing
4.7 Gallai Type Theorems
4.8 Domination and Matching
Chapter 5 Domination in Trees
5.1 Introduction
5.2 Domination in Trees
5.2.1 Domination Bounds in Trees
5.2.2 Domination Lower Bounds Involving the Number of Leaves
5.2.3 Slater Lower Bound on the Domination Number
5.2.4 Vertices in All or No Minimum Dominating Sets
5.2.5 Domination and Packing in Trees
5.3 Total Domination in Trees
5.3.1 Total Domination Bounds in Trees
5.3.2 Total Domination Bounds Involving the Number of Leaves
5.3.3 Vertices in All or No Minimum Total Dominating Sets in Trees
5.3.4 Unique Minimum Total Dominating Sets in Trees
5.3.5 Total Domination and Open Packing in Trees
5.4 Independent Domination in Trees
5.4.1 Independent Domination Bounds in Trees
5.4.2 Unique Minimum Independent Dominating Sets in Trees
5.5 Equality of Domination Parameters
5.5.1 (gamma, i)-trees
5.5.2 (gamma, gamma-t)-trees
5.5.3 Summary
Chapter 6 Upper Bounds in Terms of Minimum Degree
6.1 Introduction
6.2 Bounds on the Domination Number
6.2.1 Minimum Degree One
6.2.2 Minimum Degree Two
6.2.3 Minimum Degree Three
6.2.4 Minimum Degree Four
6.2.5 Minimum Degree Five
6.2.6 Minimum Degree Six
6.2.7 Minimum Degree Seven or Larger
6.3 Bounds on the Total Domination Number
6.3.1 Minimum Degree One
6.3.2 Minimum Degree Two
6.3.3 Minimum Degree Three
6.3.4 An Interplay with Transversals in Hypergraphs
6.3.5 Minimum Degree Four
6.3.6 Minimum Degree Five
6.3.7 Minimum Degree Six
6.3.8 A Heuristic Bound
6.4 Bounds on the Independent Domination Number
6.4.1 Minimum Degree One
6.4.2 Arbitrary Minimum Degree
6.4.3 Regular Graphs
6.5 Summary
Chapter 7 Probabilistic Bounds and Domination in Random Graphs
7.1 Introduction
7.2 Probabilistic Bounds
7.3 Domination in Random Graphs
7.4 Summary
Chapter 8 Bounds in Terms of Size
8.1 Introduction
8.2 Domination and Size
8.3 Total Domination and Size
8.4 Independent Domination and Size
8.5 Summary
Chapter 9 Efficient Domination in Graphs
9.1 Introduction
9.1.1 Efficient Dominating Sets
9.1.2 Efficient Total Dominating Sets
9.1.3 Perfect Dominating Sets
9.1.4 Examples
9.2 Efficient Domination
9.2.1 Efficient Graphs
9.2.2 Efficient Grid Graphs and Efficient Toroidal Graphs
9.2.3 Efficient Cube-connected Cycles
9.2.4 Efficient Vertex-transitive Graphs
9.2.5 Efficient Cayley Graphs
9.2.6 Efficient Circulant Graphs
9.2.7 Efficient Graphs with Efficient Complements
9.3 Efficient Total Domination
9.3.1 Total Efficient Trees
9.3.2 Total Efficient Grid Graphs
9.3.3 Total Efficient Cylindrical Graphs
9.3.4 Total Efficient Toroidal Graphs
9.3.5 Total Efficient Product Graphs
9.3.6 Total Efficient Circulant Graphs
9.4 Algorithms and Complexity of Efficient Domination
Chapter 10 Domination and Forbidden Subgraphs
10.1 Introduction
10.2 Domination and Forbidden Cycles
10.2.1 Domination Number
Forbidden 4- and 5-Cycles and Minimum Degree Two
Domination and Large Girth
10.2.2 Total Domination Number
Forbidden Induced 6-cycles and Minimum Degree Two
Forbidden 4- and 6-cycles and Minimum Degree Three
Forbidden 4-cycles and Minimum Degree Four
Total Domination and Large Girth
10.2.3 Independent Domination Number
Forbidden 4-cycles
Bipartite Graphs
Triangle-free Graphs
10.3 Domination in Claw-free Graphs
10.3.1 Domination and Independent Domination Numbers
10.3.2 Total Domination Number
10.4 Summary
Chapter 11 Domination in Planar Graphs
11.1 Introduction
11.2 Domination in Planar Graphs
11.2.1 Domination in Planar Triangulations
11.2.2 Domination in Outerplanar Graphs
11.2.3 Domination in Planar Graphs with Small Diameter
11.3 Total Domination in Planar Graphs
11.3.1 Total Domination in Outerplanar Graphs
11.3.2 Total Domination in Planar Graphs with Small Diameter
11.4 Independent Domination in Planar Graphs
Chapter 12 Domination Partitions
12.1 Introduction
12.2 Domatic Numbers
12.2.1 Domatically Full Graphs
12.2.2 Lower Bounds
12.2.3 Generalizations of the Domatic Number
Transitivity and Upper Domatic Number
Coalition Partitions
12.3 Idomatic Number
12.4 Total Domatic Number
12.4.1 Total Domatic Number in Graph Families
12.4.2 Total Domatic Number in Planar Graphs
12.5 Results of Zelinka on Domatic Numbers
12.6 Dominating Bipartitions of Graphs
12.6.1 Dominating and Total Dominating Set Partition
12.6.2 Total Dominating Set and Independent Dominating Set Partition
12.6.3 Partitions into Two Total Dominating Sets
Chapter 13 Domination Critical and Stable Graphs
13.1 Introduction
13.2 The Six Graph Families
13.2.1 CVR, CER, and CEA
13.2.2 UVR, UER, and UEA
13.2.3 Relationships Among the Families
13.3 Domination Vertex-Critical Graphs (CVR)
13.3.1 Vertex-Critical Graphs
13.3.2 3-Vertex-Critical Graphs
13.4 Domination Edge-Critical Graphs (CEA)
13.4.1 Properties of k-Edge-Critical Graphs
13.4.2 3-Edge-Critical Graphs
13.5 Total Domination Edge-Critical Graphs
13.5.1 k-t-Edge-Critical Graphs
13.5.2 3-t-Edge-Critical Graphs
Chapter 14 Upper Domination Parameters
14.1 Introduction
14.2 Upper Bounds
14.2.1 Upper Bounds in Terms of Minimum Degree
14.2.2 Upper Bounds in Regular Graphs
14.2.3 Upper Bounds in Claw-free Graphs
14.3 Upper Domination Number
14.4 Independence Number
Chapter 15 Relating the Core Parameters
15.1 Introduction
15.2 Well-covered andWell-dominated Graphs
15.2.1 Well-covered Graphs
15.2.2 Well-dominated Graphs
15.2.3 Well-total Dominated Graphs
15.3 Domination Versus Independent Domination
15.3.1 (gamma, i)-graphs
15.3.2 Domination Perfect Graphs
15.3.3 Regular Graphs
15.4 Domination Versus Total Domination
15.4.1 Regular Graphs
15.4.2 Claw-free Graphs
15.5 Upper Domination Versus Independence
15.6 Upper Domination Versus Upper Total Domination
15.6.1 Regular Graphs
15.7 Independence Versus Total Domination
15.7.1 Independent Domination Versus Total Domination
15.7.2 Independence Versus Total Domination
15.8 Summary
Chapter 16 Nordhaus-Gaddum Bounds
16.1 Introduction
16.2 Domination Number
16.2.1 Minimum Degree One
16.2.2 Minimum Degree Two
16.2.3 Minimum Degree Three
16.2.4 Minimum Degree Four
16.2.5 Minimum Degree Five
16.2.6 Minimum Degree Six
16.2.7 Summary of Bounds with Specified Minimum Degree
16.2.8 Multiple Factors
16.2.9 Relative Complement
16.3 Total Domination Number
16.4 Independent Domination Number
16.5 Upper Domination Parameters
16.5.1 Upper Domination and Independence Numbers
16.5.2 Upper Total Domination Number
16.6 Domatic Numbers of G and Ḡ
Chapter 17 Domination in Grids and Hypercubes
17.1 Introduction
17.2 Domination in Grids
17.2.1 Domination Numbers of Grids
17.2.2 Independent Domination Numbers of Grids
17.2.3 Total Domination Numbers of Grids
17.3 Domination in Hypercubes
Chapter 18 Domination and Vizing’s Conjecture
18.1 Introduction
18.2 Vizing’s Conjecture for the Domination Number
18.2.1 A Framework
18.2.2 Key Preliminary Lemmas
18.2.3 Classical Results Related to Vizing’s Conjecture
18.3 Total Domination Number
18.4 Independent Domination Number
18.5 Independence Number
18.6 Upper Domination Number
18.7 Upper Total Domination Number
Epilogue
Appendix A Glossary
A.1 Introduction
A.2 Basic Graph Theory Definitions
A.2.1 Basic Numbers
A.2.2 Common Types of Graphs
A.2.3 Graph Constructions
A.3 Graph Parameters
A.3.1 Connectivity and Subgraph Numbers
A.3.2 Distance Numbers
A.3.3 Covering, Packing, Independence, and Matching Numbers
A.3.4 Core Domination Numbers
A.3.5 Domatic Partitions
A.3.6 Perfect and Efficient Dominating Sets
A.3.7 Enclaveless Sets
A.3.8 Grid Graphs
A.4 Hypergraph Terminology and Concepts
Appendix B Books Containing Information on Domination in Graphs
Appendix C Surveys Containing Information on Domination in Graphs
Bibliography
Index
Symbol index
Subject index
Author index

Citation preview

Springer Monographs in Mathematics

Teresa W. Haynes Stephen T. Hedetniemi Michael A. Henning

Domination in Graphs: Core Concepts

Springer Monographs in Mathematics Editor-in-Chief Minhyong Kim, School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea, International Centre for Mathematical Sciences, Edinburgh, UK Katrin Wendland, School of Mathematics, Trinity College Dublin, Dublin, Ireland Series Editors Sheldon Axler, Department of Mathematics, San Francisco State University, San Francisco, CA, USA Mark Braverman, Department of Mathematics, Princeton University, Princeton, NJ, USA Maria Chudnovsky, Department of Mathematics, Princeton University, Princeton, NJ, USA Tadahisa Funaki, Department of Mathematics, University of Tokyo, Tokyo, Japan Isabelle Gallagher, Département de Mathématiques et Applications, Ecole Normale Supérieure, Paris, France Sinan Güntürk, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA Claude Le Bris, CERMICS, Ecole des Ponts ParisTech, Marne la Vallée, France Pascal Massart, Département de Mathématiques, Université de Paris-Sud, Orsay, France Alberto A. Pinto, Department of Mathematics, University of Porto, Porto, Portugal Gabriella Pinzari, Department of Mathematics, University of Padova, Padova, Italy Ken Ribet, Department of Mathematics, University of California, Berkeley, CA, USA René Schilling, Institute for Mathematical Stochastics, Technical University Dresden, Dresden, Germany Panagiotis Souganidis, Department of Mathematics, University of Chicago, Chicago, IL, USA Endre Süli, Mathematical Institute, University of Oxford, Oxford, UK Shmuel Weinberger, Department of Mathematics, University of Chicago, Chicago, IL, USA Boris Zilber, Mathematical Institute, University of Oxford, Oxford, UK

This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

Teresa W. Haynes • Stephen T. Hedetniemi Michael A. Henning

Domination in Graphs: Core Concepts

Teresa W. Haynes Department of Mathematics and Statistics East Tennessee State University Johnson City, TN, USA

Stephen T. Hedetniemi School of Computing Clemson University Clemson, SC, USA

Department of Mathematics and Applied Mathematics University of Johannesburg Johannesburg, South Africa

Michael A. Henning Department of Mathematics and Applied Mathematics University of Johannesburg Johannesburg, South Africa

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-031-09496-5 (eBook) ISBN 978-3-031-09495-8 https://doi.org/10.1007/978-3-031-09496-5 © Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedication The authors dedicate this book to their coauthor and long-time good friend Pete Slater, who coauthored the predecessor of this volume, Fundamentals of Domination in Graphs, with Teresa and Steve.

Peter J. Slater (1946–2016)

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Teresa pays a special tribute to Ulysses Grant (Lit) Haynes. I love you, Dad. Steve pays a special tribute to Gerd H. Fricke (1946–2016), an exceptional mathematician and graph theorist, who made special contributions to domination in graphs, and in particular to the understanding of the concept of irredundance in graphs. Steve also pays a special tribute to Sandee, his wife of 43 years, with whom he has coauthored 78 papers. Michael pays a special tribute to his brother Paul Henning, for the countless lives he has saved as an Emergency Medicine Physician over the years.

Preface While concepts related to domination in graphs can be traced back to the Roman Empire in the fourth century AD and to the mid-1800s in connection with various chessboard problems, the mathematical concept of domination in graphs was first suggested by Kőnig in 1936, and then defined as a graph theoretical parameter by Berge in 1958. Domination in graphs experienced rapid growth from its introduction, resulting in over 1200 papers published on domination in graphs by the late 1990s. Much of the interest in domination theory in graphs is due to its applications in many areas of study, such as genetics, chemistry, computer communication networks, facility location, social networking, surveying, transporting hazardous materials, monitoring electrical power networks, school bus routing, voting, and several areas of mathematics, to name a few. Noting the need for a comprehensive survey of the literature on domination in graphs, in 1998 Haynes, Hedetniemi, and Slater published the first two books on domination, writing Fundamentals of Domination in Graphs (ISBN: 9780429157769) and editing Domination in Graphs: Advanced Topics (ISBN: 9781315141428). The explosive growth of this field has continued since 1998, and today more than 5000 papers have been published on domination in graphs, and the material in these two books is now more than 20 years old. Thus, we thought it was time for an update on the developments in domination theory since 1998. We also wanted to give a comprehensive treatment of only the major topics in domination. This coverage of domination, including the major results and updates, is in the form of three books: this book and its two companion books, Topics in Domination in Graphs (ISBN: 9783030511173) and Structures of Domination in Graphs (ISBN: 9783030588915), which we will call Books I, II, and III, respectively. This book, Domination in Graphs: Core Concepts, is limited to, as the title suggests, the most core concepts of domination in graphs: domination, total domination, and independent domination. It contains major results on these three types of domination, including a wide variety of proofs, both short and long, of selected results that illustrate many of the proof techniques used in domination theory. For the companion books, Books II and III, we invited leading researchers in domination theory to contribute chapters. Book II focuses on the most-studied types of domination that are not covered in Book I. Although well over 70 types of domination have been defined, Book II focuses on those that have received the most attention in the literature, and contains chapters on paired domination, connected domination, restrained domination, multiple domination, distance domination, dominating functions, fractional domination, Roman domination, rainbow domination, locating-domination, eternal and secure domination, global domination, stratified domination, and power domination. Book III is divided into three parts. The first part covers several domination-related concepts: broadcast domination, alliances, domatic numbers, dominator colorings, irredundance in graphs, private neighbor concepts, game domination, varieties of Roman domination, and domination in spectral graph theory. The second part contains chapters on domination in hypergraphs, chessboards, and digraphs and tournaments. vii

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The third part focuses on the development of algorithms and complexity of signed, minus, and majority domination, power domination, and alliances in graphs. The third part also includes a chapter on self-stabilizing algorithms for domination. This book (Book I) is intended as a reference resource for researchers and is written to reach the following audiences: first, established researchers in the field of domination who want an updated, comprehensive coverage of domination theory; second, researchers in graph theory who wish to become acquainted with newer topics in domination, along with major developments in the field and some of the proof techniques used; and third, graduate students with interests in graph theory, who might find the theory and many real-world applications of domination of interest for master’s and doctoral theses topics. We also believe that this book provides a good basis for use in a seminar on either domination theory or domination algorithms and complexity, including the new algorithm paradigm of self-stabilizing domination algorithms. This book is intended as an in-depth introduction to domination in graphs, limited to its most core concepts of domination, total domination, and independent domination. We have therefore intentionally focused more on depth than breadth in Book I, and supplied several in-depth proofs for the reader to acquaint themselves with a tool box of proof techniques and methods with which to attack open problems in the field. We have identified many unsolved problems and open conjectures, which can be used as a launching pad for future researchers in the field. With the enormous literature that exists on domination in graphs and the dynamic nature of the subject, we were faced with the challenge of determining which topics to include and perhaps even more importantly which topics to exclude, even for the core concepts of domination, total domination, and independent domination. We have therefore been selective in the material included in this core domination book and wish to apologize in advance for omitting many important results and proofs due to space limitations. We assume that the reader is acquainted with the basic concepts of graph theory and has had some exposure to graph theory at an introductory level. However, since graph theory terminology sometimes varies, we provide a glossary as a reference source for the reader regarding terminology and notation adopted in this book. Assuming that the reader has some familiarity with graph theory, this book is selfcontained as we include the terminology and definitions involving domination in the glossary in Appendix A. The material in this book has been organized into 18 chapters, an epilogue, and three appendices. It contains an extensive bibliography of more than 900 references, which we have cited throughout the book. A brief summary of the material covered in each chapter is presented below. Chapter 1 In the Beginning: Roots of Domination in Graphs discusses the many origins, both historical and mathematical, of domination in graphs, dating as far back as the Roman Empire in the fourth century AD under Emperor Constantine. Chapter 2 Fundamentals of Domination discusses how it is that the domination number, total domination number, and independent domination number can be defined

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in a variety of equivalent ways, each of which suggests natural generalizations of these three types of domination. Chapter 3 Complexity and Algorithms for Domination in Graphs provides an overview of the core results on NP-completeness and algorithms for domination, total domination, and independent domination in graphs. It presents NP-completeness proofs for each type of domination, when restricted to several subclasses of graphs, and provides linear algorithms for computing each type of domination on trees. Chapter 4 General Bounds presents some of the more basic lower and upper bounds on the domination, total domination, and independent domination numbers of graphs. Chapter 5 Domination in Trees presents a wide variety of domination results for the class of trees, including lower and upper bounds, bounds in terms of the number of leaves in a tree, the Slater lower bound for trees, vertices in all or no minimum dominating sets in a tree, trees in which every vertex is a member of some minimum dominating set, trees having unique minimum dominating sets, trees in which the domination number equals the independent domination number, and trees in which the domination number equals the total domination number. Chapter 6 Upper Bounds in Terms of Minimum Degree presents results which establish upper bounds on the core domination numbers in terms of the order of a graph and the minimum degree of a vertex in the graph, where for the domination number and total domination number the minimum degree ranges between one and six. Chapter 7 Probabilistic Bounds and Domination in Random Graphs presents probabilistic bounds for the core domination numbers of a graph in terms of its order and minimum degree, and also considers bounds for the domination numbers of random graphs. It covers the basic questions of the probability that a randomly chosen set 𝑆 of vertices in a graph 𝐺 is a dominating set of one of the three basic types, if each vertex in the graph is chosen to be in the set 𝑆 with a given probability. Chapter 8 Bounds in Terms of Size discusses how the number of edges of a graph affects the values of the core domination numbers. Chapter 9 Efficient Domination in Graphs considers classes of graphs that have dominating or total dominating sets 𝑆 in which specified vertices are adjacent to exactly one vertex in 𝑆. Included in these classes of graphs are certain circulants, Cayley graphs, grid graphs, cylindrical graphs, toroidal graphs, prisms, Möbius ladders, and lexicographic graphs. Also included is a section on NP-completeness results for graphs having an efficient dominating set. Chapter 10 Domination and Forbidden Subgraphs presents bounds on the three core domination numbers in classes of graphs which have certain subgraph restrictions, such as bipartite (no odd cycles), cubic (every vertex has degree three), and claw-free (no vertex has three neighbors, no two of which are adjacent). Chapter 11 Domination in Planar Graphs covers domination and total domination in planar triangulations, outerplanar graphs, and in planar graphs having small diameter. Results on independent domination in planar graphs are also presented, including bipartite, cubic, and minimum diameter planar graphs.

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Chapter 12 Domination Partitions covers vertex partitions 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } of a graph 𝐺 such that each set 𝑉𝑖 is a dominating set. Partitions into total and independent dominating sets are also considered. The nine possible ways of partitioning the vertices of a graph into two sets, say 𝑉1 and 𝑉2 , such that 𝑉1 is one of the three types of domination and 𝑉2 is one of the three types of domination are also considered. Chapter 13 Domination Critical and Stable Graphs presents the classes of graphs whose types of domination numbers change upon the removal of any vertex, the removal of any edge, or the addition of any edge. It also considers the classes of graphs whose domination numbers do not change, regardless of which vertex or edge is removed or which new edge is added to the graph. Chapter 14 Upper Domination Parameters covers the upper domination number, the upper total domination number, and the independence number, that is, the maximum cardinalities of a minimal type of dominating set. Since the independence number, that is, the maximum cardinality of an independent set, is very well-studied in the literature, the focus of this chapter is mainly on the upper domination and upper total domination numbers, although several important results on the independence number are presented. Chapter 15 Relating the Core Parameters presents relationships, inequalities, and bounds that exist between the three types of domination numbers, for example bounds on the ratio of the independent domination number to the domination number and the total domination number to the domination number. Also considered are the classes of graphs in which two of these domination numbers are always equal, for example the classes of graphs in which the independence number equals the upper domination number. Chapter 16 Nordhaus-Gaddum Bounds discusses bounds on the sum and product of the domination numbers of a graph 𝐺 and its complement 𝐺. Bounds on the sum and product for total domination and independent domination numbers are also presented. Chapter 17 Domination in Grids and Hypercubes presents results on domination, total domination, and independent domination in grids, which are chessboard-like graphs. There is also a brief discussion of cylinders (chessboards with column wrap-arounds) and tori (chessboards with both column and row wrap-arounds). The chapter concludes with domination in hypercubes. Chapter 18 Domination and Vizing’s Conjecture provides an overview of the most well-known conjecture in domination theory, that the domination number of the Cartesian product of two graphs 𝐺 and 𝐻 is greater than or equal to the product of the domination number of 𝐺 and the domination number of 𝐻. Similar conjectures are also discussed for all six core domination numbers, including the lower and upper domination, total domination, and independent domination numbers. The authors would like to thank Elizabeth Loew, the Executive Editor, Mathematics at Springer, and Saveetha Balasundaram, the Project Coordinator (Books) for Springer Nature, for their continued support and encouragement, not only in producing this book but throughout the production of Books II and III. We are especially grateful to them for their patience in waiting for this manuscript from the date the contract was signed, and for their cooperation in all aspects of the production of this book.

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The authors thank Jason Hedetniemi for proofreading several chapters of this book. In addition, the authors thank the five Springer reviewers of an early version of this book for their many helpful suggestions, which lead to a much improved book. We extend a hearty and sincere thanks to Dr Werner Gründlingh for his tireless and outstanding efforts in typesetting the book and for producing superb graphics. We very much appreciate his expertise and skills, and the enormous time sacrifice he has made in assisting us during the typesetting process. Teresa Haynes would like to thank East Tennessee State University and the University of Johannesburg for their support during the writing of this book. In particular, she extends a special acknowledgement to the staff at Sherrod Library for their assistance. She also thanks Pamela Morgan for her friendship and helpful encouragement for this project. Stephen Hedetniemi would like to thank Clemson University and the Clemson University Library, the School of Computing, and the Emeritus College for their support in producing Books I, II, and III. He also thanks his wife, Sandee, and children Traci, Laura, Kevin, and Jason for their encouragement in writing this book. Michael Henning expresses his sincere thanks to the University of Johannesburg for their continued support and for creating a conducive research environment to enable him to work on the book. Special thanks are due to his wife Anne, son John, and daughter Alicen, for their much appreciated support and encouragement throughout the writing of the book. We have tried to eliminate errors, but surely several remain. We welcome any comments or corrections the reader may have. A list of typographical errors, corrections, and suggestions can be sent to any of our email addresses below. East Tennessee State University, USA and University of Johannesburg, South Africa Clemson University, USA University of Johannesburg, South Africa

Teresa W. Haynes e-mail: [email protected] Stephen T. Hedetniemi e-mail: [email protected] Michael A. Henning e-mail: [email protected]

Contents 1 In the Beginning: Roots of Domination in Graphs 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Basic Terminology . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Defensive and Offensive Strategies of the Roman Empire 1.3.2 Chaturanga . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Eight Queens Problem . . . . . . . . . . . . . . . . . . 1.3.4 Five Queens Problem . . . . . . . . . . . . . . . . . . . 1.3.5 Queens Independent Domination Problem . . . . . . . . 1.3.6 Queens Total Domination Problem . . . . . . . . . . . . 1.3.7 Generalizations to Other Chess Pieces . . . . . . . . . . 1.4 Application Driven Origins . . . . . . . . . . . . . . . . . . . . 1.4.1 Radio Broadcasting . . . . . . . . . . . . . . . . . . . . 1.4.2 Computer Communication Networks . . . . . . . . . . 1.4.3 Sets of Representatives . . . . . . . . . . . . . . . . . . 1.4.4 School Bus Routing and Bus Stop Selection . . . . . . . 1.4.5 Electrical Power Domination . . . . . . . . . . . . . . . 1.4.6 Influence in Social Networks . . . . . . . . . . . . . . . 1.4.7 Topographic Maps . . . . . . . . . . . . . . . . . . . . 1.4.8 Transporting Hazardous Materials . . . . . . . . . . . . 1.5 Early Chronology of Domination in Graph Theory . . . . . . . . 2

Fundamentals of Domination 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Core Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Independent Sets . . . . . . . . . . . . . . . . . . . 2.2.2 Dominating Sets . . . . . . . . . . . . . . . . . . . 2.2.3 Irredundant Sets . . . . . . . . . . . . . . . . . . . 2.3 Parameters Suggested by the Definition of a Dominating Set 2.3.1 Total Dominating Sets . . . . . . . . . . . . . . . . 2.3.2 𝑘-Dominating Sets . . . . . . . . . . . . . . . . . . 2.3.3 𝐻-forming Dominating Sets . . . . . . . . . . . . . 2.3.4 Perfect and Efficient Dominating Sets . . . . . . . .

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2.3.5 Distance-𝑘 Dominating Sets . . . . . . 2.3.6 Fractional Domination . . . . . . . . . Equivalent Formulations of Domination . . . . 2.4.1 Pendant Edges in Spanning Forests . . 2.4.2 Enclaveless Sets . . . . . . . . . . . . 2.4.3 Spanning Star Partitions . . . . . . . . 2.4.4 Non-dominating Partitions . . . . . . . 2.4.5 Total Domination and Splitting Graphs 2.4.6 Dominating Sets and (1, 4 : 3)-Sets . . . Domination in Terms of Perfection in Graphs . Ore’s Lemmas and Their Implications . . . . .

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Complexity and Algorithms for Domination in Graphs 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Brief Review of NP-Completeness . . . . . . . . . . . . . . . . 3.3 NP-Completeness of Domination, Independent Domination, and Total Domination . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 NP-Completeness Results for Arbitrary Graphs . . . . . 3.3.2 NP-Completeness Results for Bipartite Graphs . . . . . 3.3.3 Summary of Complexity Results for Graph Families . . 3.4 A Representative Sample of Domination Algorithms for Trees . 3.4.1 Minimum Dominating Set . . . . . . . . . . . . . . . . 3.4.2 Minimum Independent Dominating Set . . . . . . . . . 3.4.3 Minimum Total Dominating Set . . . . . . . . . . . . . 3.5 Early Domination Algorithms and NP-Completeness Results . . 3.6 Other Sources for Domination Algorithms and Complexity . . .

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52 52 56 58 58 59 62 63 65 70

General Bounds 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Domination and Maximum Degree . . . . . . . . . . . . . . . . 4.2.1 Domination Number and Maximum Degree . . . . . . . 4.2.2 Total Domination Number and Maximum Degree . . . . 4.2.3 Independent Domination Number and Maximum Degree 4.3 Domination and Order . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Domination Number and Order . . . . . . . . . . . . . 4.3.2 Total Domination Number and Order . . . . . . . . . . 4.3.3 Independent Domination Number and Order . . . . . . . 4.4 Basic Relationships Among Core Parameters . . . . . . . . . . 4.5 Domination and Distance . . . . . . . . . . . . . . . . . . . . . 4.6 Domination and Packing . . . . . . . . . . . . . . . . . . . . . 4.7 Gallai Type Theorems . . . . . . . . . . . . . . . . . . . . . . . 4.8 Domination and Matching . . . . . . . . . . . . . . . . . . . .

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73 73 74 74 76 79 79 80 82 86 87 88 91 92 94

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5 Domination in Trees 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Domination in Trees . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Domination Bounds in Trees . . . . . . . . . . . . . . . . . 5.2.2 Domination Lower Bounds Involving the Number of Leaves 5.2.3 Slater Lower Bound on the Domination Number . . . . . . 5.2.4 Vertices in All or No Minimum Dominating Sets . . . . . . 5.2.5 Domination and Packing in Trees . . . . . . . . . . . . . . 5.3 Total Domination in Trees . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Total Domination Bounds in Trees . . . . . . . . . . . . . . 5.3.2 Total Domination Bounds Involving the Number of Leaves . 5.3.3 Vertices in All or No Minimum Total Dominating Sets in Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Unique Minimum Total Dominating Sets in Trees . . . . . . 5.3.5 Total Domination and Open Packing in Trees . . . . . . . . 5.4 Independent Domination in Trees . . . . . . . . . . . . . . . . . . . 5.4.1 Independent Domination Bounds in Trees . . . . . . . . . . 5.4.2 Unique Minimum Independent Dominating Sets in Trees . . 5.5 Equality of Domination Parameters . . . . . . . . . . . . . . . . . . 5.5.1 (𝛾, 𝑖)-trees . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 (𝛾, 𝛾t )-trees . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 99 100 100 103 109 112 114 115 115 116

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Upper Bounds in Terms of Minimum Degree 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 6.2 Bounds on the Domination Number . . . . . . . . . . 6.2.1 Minimum Degree One . . . . . . . . . . . . . 6.2.2 Minimum Degree Two . . . . . . . . . . . . . 6.2.3 Minimum Degree Three . . . . . . . . . . . . 6.2.4 Minimum Degree Four . . . . . . . . . . . . . 6.2.5 Minimum Degree Five . . . . . . . . . . . . . 6.2.6 Minimum Degree Six . . . . . . . . . . . . . . 6.2.7 Minimum Degree Seven or Larger . . . . . . . 6.3 Bounds on the Total Domination Number . . . . . . . 6.3.1 Minimum Degree One . . . . . . . . . . . . . 6.3.2 Minimum Degree Two . . . . . . . . . . . . . 6.3.3 Minimum Degree Three . . . . . . . . . . . . 6.3.4 An Interplay with Transversals in Hypergraphs 6.3.5 Minimum Degree Four . . . . . . . . . . . . . 6.3.6 Minimum Degree Five . . . . . . . . . . . . . 6.3.7 Minimum Degree Six . . . . . . . . . . . . . . 6.3.8 A Heuristic Bound . . . . . . . . . . . . . . . 6.4 Bounds on the Independent Domination Number . . . 6.4.1 Minimum Degree One . . . . . . . . . . . . . 6.4.2 Arbitrary Minimum Degree . . . . . . . . . .

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6.4.3 Regular Graphs . . . . . . . . . . . . . . . . . . . . . . . . 200 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

7 Probabilistic Bounds and Domination in Random Graphs 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 7.2 Probabilistic Bounds . . . . . . . . . . . . . . . . . . 7.3 Domination in Random Graphs . . . . . . . . . . . . . 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . .

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227 227 227 238 255 258

9 Efficient Domination in Graphs 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Efficient Dominating Sets . . . . . . . . . . . . . 9.1.2 Efficient Total Dominating Sets . . . . . . . . . . 9.1.3 Perfect Dominating Sets . . . . . . . . . . . . . . 9.1.4 Examples . . . . . . . . . . . . . . . . . . . . . . 9.2 Efficient Domination . . . . . . . . . . . . . . . . . . . . 9.2.1 Efficient Graphs . . . . . . . . . . . . . . . . . . 9.2.2 Efficient Grid Graphs and Efficient Toroidal Graphs 9.2.3 Efficient Cube-connected Cycles . . . . . . . . . . 9.2.4 Efficient Vertex-transitive Graphs . . . . . . . . . 9.2.5 Efficient Cayley Graphs . . . . . . . . . . . . . . 9.2.6 Efficient Circulant Graphs . . . . . . . . . . . . . 9.2.7 Efficient Graphs with Efficient Complements . . . 9.3 Efficient Total Domination . . . . . . . . . . . . . . . . . 9.3.1 Total Efficient Trees . . . . . . . . . . . . . . . . 9.3.2 Total Efficient Grid Graphs . . . . . . . . . . . . . 9.3.3 Total Efficient Cylindrical Graphs . . . . . . . . . 9.3.4 Total Efficient Toroidal Graphs . . . . . . . . . . . 9.3.5 Total Efficient Product Graphs . . . . . . . . . . . 9.3.6 Total Efficient Circulant Graphs . . . . . . . . . . 9.4 Algorithms and Complexity of Efficient Domination . . . .

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259 259 259 260 261 261 262 263 264 266 267 268 270 272 272 273 274 275 276 280 282 282

10 Domination and Forbidden Subgraphs 10.1 Introduction . . . . . . . . . . . . . . . . 10.2 Domination and Forbidden Cycles . . . . 10.2.1 Domination Number . . . . . . . 10.2.2 Total Domination Number . . . . 10.2.3 Independent Domination Number 10.3 Domination in Claw-free Graphs . . . . .

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291 291 291 291 299 310 314

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Bounds in Terms of Size 8.1 Introduction . . . . . . . . . . . . 8.2 Domination and Size . . . . . . . 8.3 Total Domination and Size . . . . 8.4 Independent Domination and Size 8.5 Summary . . . . . . . . . . . . .

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10.3.1 Domination and Independent Domination Numbers . . . . . 314 10.3.2 Total Domination Number . . . . . . . . . . . . . . . . . . 318 10.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 11 Domination in Planar Graphs 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Domination in Planar Graphs . . . . . . . . . . . . . . . . . . . 11.2.1 Domination in Planar Triangulations . . . . . . . . . . . 11.2.2 Domination in Outerplanar Graphs . . . . . . . . . . . . 11.2.3 Domination in Planar Graphs with Small Diameter . . . 11.3 Total Domination in Planar Graphs . . . . . . . . . . . . . . . . 11.3.1 Total Domination in Outerplanar Graphs . . . . . . . . 11.3.2 Total Domination in Planar Graphs with Small Diameter 11.4 Independent Domination in Planar Graphs . . . . . . . . . . . .

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325 325 326 326 333 337 341 341 344 347

12 Domination Partitions 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 12.2 Domatic Numbers . . . . . . . . . . . . . . . . . . . . 12.2.1 Domatically Full Graphs . . . . . . . . . . . . 12.2.2 Lower Bounds . . . . . . . . . . . . . . . . . 12.2.3 Generalizations of the Domatic Number . . . . 12.3 Idomatic Number . . . . . . . . . . . . . . . . . . . . 12.4 Total Domatic Number . . . . . . . . . . . . . . . . . 12.4.1 Total Domatic Number in Graph Families . . . 12.4.2 Total Domatic Number in Planar Graphs . . . . 12.5 Results of Zelinka on Domatic Numbers . . . . . . . . 12.6 Dominating Bipartitions of Graphs . . . . . . . . . . . 12.6.1 Dominating and Total Dominating Set Partition 12.6.2 Total and Independent Dominating Set Partition 12.6.3 Partitions into Two Total Dominating Sets . . .

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353 353 353 356 358 359 361 363 365 365 366 368 368 370 370

13 Domination Critical and Stable Graphs 13.1 Introduction . . . . . . . . . . . . . . . . . . 13.2 The Six Graph Families . . . . . . . . . . . . 13.2.1 CVR, CER, and CEA . . . . . . . . . 13.2.2 UVR, UER, and UEA . . . . . . . . 13.2.3 Relationships Among the Families . . 13.3 Domination Vertex-Critical Graphs (CVR) . . 13.3.1 Vertex-Critical Graphs . . . . . . . . 13.3.2 3-Vertex-Critical Graphs . . . . . . . 13.4 Domination Edge-Critical Graphs (CEA) . . 13.4.1 Properties of 𝑘-Edge-Critical Graphs 13.4.2 3-Edge-Critical Graphs . . . . . . . . 13.5 Total Domination Edge-Critical Graphs . . . 13.5.1 𝑘 t -Edge-Critical Graphs . . . . . . .

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13.5.2 3t -Edge-Critical Graphs . . . . . . . . . . . . . . . . . . . 403

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411 411 412 413 414 419 426 428

15 Relating the Core Parameters 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Well-covered and Well-dominated Graphs . . . . . . . . . 15.2.1 Well-covered Graphs . . . . . . . . . . . . . . . . 15.2.2 Well-dominated Graphs . . . . . . . . . . . . . . 15.2.3 Well-total Dominated Graphs . . . . . . . . . . . 15.3 Domination Versus Independent Domination . . . . . . . . 15.3.1 (𝛾, 𝑖)-graphs . . . . . . . . . . . . . . . . . . . . 15.3.2 Domination Perfect Graphs . . . . . . . . . . . . . 15.3.3 Regular Graphs . . . . . . . . . . . . . . . . . . . 15.4 Domination Versus Total Domination . . . . . . . . . . . 15.4.1 Regular Graphs . . . . . . . . . . . . . . . . . . . 15.4.2 Claw-free Graphs . . . . . . . . . . . . . . . . . . 15.5 Upper Domination Versus Independence . . . . . . . . . . 15.6 Upper Domination Versus Upper Total Domination . . . . 15.6.1 Regular Graphs . . . . . . . . . . . . . . . . . . . 15.7 Independence Versus Total Domination . . . . . . . . . . 15.7.1 Independent Domination Versus Total Domination 15.7.2 Independence Versus Total Domination . . . . . . 15.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .

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435 435 436 437 438 440 440 441 442 445 451 451 454 454 458 461 463 463 465 465

16 Nordhaus-Gaddum Bounds 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Domination Number . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Minimum Degree One . . . . . . . . . . . . . . . . . 16.2.2 Minimum Degree Two . . . . . . . . . . . . . . . . . 16.2.3 Minimum Degree Three . . . . . . . . . . . . . . . . 16.2.4 Minimum Degree Four . . . . . . . . . . . . . . . . . 16.2.5 Minimum Degree Five . . . . . . . . . . . . . . . . . 16.2.6 Minimum Degree Six . . . . . . . . . . . . . . . . . . 16.2.7 Summary of Bounds with Specified Minimum Degree 16.2.8 Multiple Factors . . . . . . . . . . . . . . . . . . . . 16.2.9 Relative Complement . . . . . . . . . . . . . . . . . . 16.3 Total Domination Number . . . . . . . . . . . . . . . . . . .

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467 467 467 470 474 476 478 478 479 480 480 484 485

14 Upper Domination Parameters 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 14.2 Upper Bounds . . . . . . . . . . . . . . . . . . . . . 14.2.1 Upper Bounds in Terms of Minimum Degree 14.2.2 Upper Bounds in Regular Graphs . . . . . . 14.2.3 Upper Bounds in Claw-free Graphs . . . . . 14.3 Upper Domination Number . . . . . . . . . . . . . . 14.4 Independence Number . . . . . . . . . . . . . . . .

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16.4 Independent Domination Number . . . . . . . . . . . 16.5 Upper Domination Parameters . . . . . . . . . . . . . 16.5.1 Upper Domination and Independence Numbers 16.5.2 Upper Total Domination Number . . . . . . . 16.6 Domatic Numbers of 𝐺 and 𝐺 . . . . . . . . . . . . .

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505 505 505 506 510 512 520

18 Domination and Vizing’s Conjecture 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Vizing’s Conjecture for the Domination Number . . . . . 18.2.1 A Framework . . . . . . . . . . . . . . . . . . . 18.2.2 Key Preliminary Lemmas . . . . . . . . . . . . 18.2.3 Classical Results Related to Vizing’s Conjecture 18.3 Total Domination Number . . . . . . . . . . . . . . . . 18.4 Independent Domination Number . . . . . . . . . . . . 18.5 Independence Number . . . . . . . . . . . . . . . . . . 18.6 Upper Domination Number . . . . . . . . . . . . . . . . 18.7 Upper Total Domination Number . . . . . . . . . . . . .

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525 525 525 527 531 533 537 539 540 541 546

17 Domination in Grids and Hypercubes 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . 17.2 Domination in Grids . . . . . . . . . . . . . . . . 17.2.1 Domination Numbers of Grids . . . . . . . 17.2.2 Independent Domination Numbers of Grids 17.2.3 Total Domination Numbers of Grids . . . . 17.3 Domination in Hypercubes . . . . . . . . . . . . .

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Epilogue A Glossary A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Basic Graph Theory Definitions . . . . . . . . . . . . . . . . . . A.2.1 Basic Numbers . . . . . . . . . . . . . . . . . . . . . . . A.2.2 Common Types of Graphs . . . . . . . . . . . . . . . . . A.2.3 Graph Constructions . . . . . . . . . . . . . . . . . . . . A.3 Graph Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 Connectivity and Subgraph Numbers . . . . . . . . . . . A.3.2 Distance Numbers . . . . . . . . . . . . . . . . . . . . . A.3.3 Covering, Packing, Independence, and Matching Numbers A.3.4 Core Domination Numbers . . . . . . . . . . . . . . . . . A.3.5 Domatic Partitions . . . . . . . . . . . . . . . . . . . . . A.3.6 Perfect and Efficient Dominating Sets . . . . . . . . . . . A.3.7 Enclaveless Sets . . . . . . . . . . . . . . . . . . . . . . A.3.8 Grid Graphs . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Hypergraph Terminology and Concepts . . . . . . . . . . . . . . B Books Containing Information on Domination in Graphs

549

. . . . . . . . . . . . . . .

555 555 555 558 558 559 560 560 561 561 563 564 564 565 565 565 567

xx

Contents

C Surveys Containing Information on Domination in Graphs

571

Bibliography

575

Index 623 Symbol index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634

Chapter 1

In the Beginning: Roots of Domination in Graphs 1.1 Introduction While domination in graphs was first formally defined by Berge in 1958, the roots of domination can be traced back to defense strategies used by the Roman Empire in the fourth century AD, to a precursor of the game of chess in India in the sixth century AD, and later in the mid-to-late 1800s, to a variety of chess problems. Other sources of domination can be found in a wide array of real-world areas such as radio broadcasting, computer communication networks, systems of distinct representatives, school bus routing, electrical power networks, influence in social networks, surveying, resource allocation, and even transporting hazardous materials. In the 1900s, a variety of international researchers began to develop the mathematical foundations of domination in graphs, including the British mathematician, lawyer, and fellow at Trinity College Cambridge, W.W. Rouse Ball; the Hungarian mathematician who wrote the first book on graph theory, Dénes Kőnig; the English mathematician and statistician, Patrick Michael Grundy; the Hungarian-American mathematician, physicist, computer scientist, and engineer, John von Neumann; the German-American economist, Oskar Morgenstern; the French mathematician recognized as one of the founders of graph theory, Claude Berge; the Hungarian graph theorist Tibor Gallai; the Norwegian-American mathematician who worked in ring theory, Galois theory, and graph theory, Øystein Ore; the Soviet and Ukrainian graph theorist, Vadim Vizing; the Finnish mathematician, Juho Nieminen; and the Canadian graph theorists, Amram Meir, John Moon, and E.J. Cockayne. In this chapter, we discuss the many origins, both historical and mathematical, of domination in graphs and highlight some of the most significant contributions of these mathematicians to the theory of domination up to the year 1998 when the first two books on domination in graphs were produced by the American graph theorists, Teresa Haynes, Stephen Hedetniemi, and Peter Slater [416, 417]. © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_1

1

2

Chapter 1. In the Beginning: Roots of Domination in Graphs

Before delving into the roots of domination in graphs, we give some basic definitions and notation in Section 1.2 that will be used throughout the book. To avoid repeating terminology in every chapter, we also provide a glossary in Appendix A including these basic terms and other definitions and refer the reader to it for terminology not defined on the spot.

1.2

Basic Terminology

A graph 𝐺 = (𝑉, 𝐸) consists of a finite nonempty set 𝑉 (𝐺) of objects called vertices together with a possibly empty set 𝐸 (𝐺) of 2-element subsets of 𝑉 (𝐺) called edges. Throughout, unless otherwise stated, the graphs in this book are simple graphs with no loops or multiple edges and 𝐺 is a graph with vertex set 𝑉 and edge set 𝐸. The number of vertices 𝑛 = |𝑉 | is called the order of 𝐺 and the number of edges 𝑚 = |𝐸 | is the size of 𝐺. An edge {𝑢, 𝑣} is denoted by 𝑢𝑣. If 𝑢𝑣 ∈ 𝐸, then 𝑢 and 𝑣 are adjacent vertices. The vertex 𝑢 (respectively, 𝑣) and edge 𝑢𝑣 are said to be incident to each other. Two distinct edges are adjacent if they are incident to a common vertex. The graph consisting of a single vertex is called the trivial graph; a nontrivial graph has order 𝑛 ≥ 2. Given a graph 𝐺 = (𝑉, 𝐸), the complement 𝐺 of 𝐺 is the graph 𝐺 = (𝑉, 𝐸), where 𝑢𝑣 ∈ 𝐸 if and only if 𝑢𝑣 ∉ 𝐸. The complete graph 𝐾𝑛 is a graph of order 𝑛 in which every two vertices are adjacent, while its complement 𝐾 𝑛 is an empty graph, that is, a graph on 𝑛 vertices with no edges. Note that 𝐾1 is the trivial graph. The open neighborhood of a vertex 𝑣 ∈ 𝑉 is the set N𝐺 (𝑣) = {𝑢 : 𝑢𝑣 ∈ 𝐸 } of vertices adjacent to 𝑣, called the neighbors of 𝑣, and its closed neighborhood is the set N𝐺 [𝑣] = N Ð𝐺 (𝑣) ∪ {𝑣}. The open neighborhood of a set 𝑆 ⊆ 𝑉 of vertices is N𝐺 (𝑆) = 𝑣 ∈𝑆 N𝐺 (𝑣), while the closed neighborhood of a set 𝑆 is Ð the set N𝐺 [𝑆] = 𝑣 ∈𝑆 N𝐺 [𝑣]. The degree of a vertex 𝑣 is deg𝐺 (𝑣) = |N𝐺 (𝑣)|. If the graph 𝐺 is clear from the context, then we omit the 𝐺 subscript in the above expressions. A vertex 𝑣 ∈ 𝑉 is called an isolated vertex if deg(𝑣) = 0, and is called a leaf if deg(𝑣) = 1. In a graph 𝐺 of order 𝑛, a vertex 𝑣 for which deg(𝑣) = 𝑛 − 1 is called a dominating vertex. A graph 𝐺 is called 𝑘-regular if every vertex 𝑣 ∈ 𝑉 has deg(𝑣) = 𝑘. We say that a graph is isolate-free if it has no isolated vertices. The largest degree among the vertices of 𝐺 is the maximum degree Δ(𝐺) and the smallest degree is the minimum degree 𝛿(𝐺). A graph 𝐺 ′ = (𝑉 ′ , 𝐸 ′ ) is a subgraph of a graph 𝐺 = (𝑉, 𝐸) if 𝑉 ′ ⊆ 𝑉 and 𝐸 ′ ⊆ 𝐸 and 𝐺 ′ is a spanning subgraph of 𝐺 if 𝑉 ′ = 𝑉. For a nonempty subset 𝑆 ⊆ 𝑉, the subgraph 𝐺 [𝑆] of 𝐺 induced by 𝑆 has 𝑆 as its vertex set and two vertices 𝑢 and 𝑣 are adjacent in 𝐺 [𝑆] if and only if 𝑢 and 𝑣 are adjacent in 𝐺. A clique is a complete subgraph. A set 𝑆 of vertices of a graph 𝐺 is a dominating set if every vertex in 𝑉 \ 𝑆 has a neighbor in 𝑆, that is, N[𝑆] = 𝑉. The domination number 𝛾(𝐺) equals the minimum cardinality of a dominating set of 𝐺 and a dominating set with cardinality 𝛾(𝐺) is called a 𝛾-set of 𝐺.

Section 1.2. Basic Terminology

3

A set 𝑆 of vertices of an isolate-free graph 𝐺 is a total dominating set, abbreviated TD-set, if every vertex in 𝑉 is adjacent to at least one vertex in 𝑆. Thus, a subset 𝑆 ⊆ 𝑉 is a TD-set of 𝐺 if N(𝑆) = 𝑉. Note that since every vertex must have a neighbor in 𝑆, total domination is only defined for isolate-free graphs. The total domination number 𝛾t (𝐺) equals the minimum cardinality of a TD-set of 𝐺 and a TD-set with cardinality 𝛾t (𝐺) is called a 𝛾t -set of 𝐺. A minimal dominating set in a graph 𝐺 is a dominating set that contains no dominating set of 𝐺 as a proper subset, and a minimal TD-set of 𝐺 is a TD-set that contains no TD-set of 𝐺 as a proper subset. The upper domination number Γ(𝐺) equals the maximum cardinality of a minimal dominating set in 𝐺. Similarly, the upper total domination number Γt (𝐺) equals the maximum cardinality of a minimal TD-set of 𝐺. A set 𝑆 ⊆ 𝑉 is independent if no two vertices in 𝑆 are adjacent in 𝐺, and an independent set 𝑆 is called maximal if no proper superset of 𝑆 is independent. The vertex independence number, or just independence number, 𝛼(𝐺) equals the maximum cardinality of an independent set of 𝐺. A set 𝑆 ⊆ 𝑉 is an independent dominating set, abbreviated ID-set, if it is both independent and dominating. The independent domination number 𝑖(𝐺) equals the minimum cardinality of any ID-set of 𝐺 and an ID-set with cardinality 𝑖(𝐺) is called an 𝑖-set of 𝐺. We note that 𝑖(𝐺) is the minimum cardinality of any maximal independent set of 𝐺. A set 𝑀 ⊆ 𝐸 is independent if no two edges in 𝑀 are adjacent in 𝐺, and a set of independent edges is called a matching. The matching number 𝛼′ (𝐺) equals the maximum number of edges in a matching of 𝐺. A set 𝑆 ⊆ 𝑉 is a packing in 𝐺 if for any two vertices 𝑢, 𝑣 ∈ 𝑆, N[𝑢] ∩ N[𝑣] = ∅. The packing number 𝜌(𝐺) equals the maximum cardinality of a packing of 𝐺. A vertex cover is a set 𝑆 of vertices such that every edge in 𝐸 is incident to at least one vertex in 𝑆. The vertex covering number 𝛽(𝐺), also denoted 𝜏(𝐺), equals the minimum cardinality of a vertex cover of 𝐺. An edge cover is a set 𝐹 of edges such that every vertex in 𝑉 is incident to at least one edge in 𝐹. The edge covering number 𝛽′ (𝐺) equals the minimum cardinality of an edge cover of 𝐺. These concepts will be explored in more detail in Chapters 2 and 4. A graph 𝐺 is bipartite if its vertex set 𝑉 can be partitioned into two sets 𝑋 and 𝑌 such that every edge in 𝐺 joins a vertex in 𝑋 and a vertex in 𝑌 . The sets 𝑋 and 𝑌 are called the partite sets of 𝐺. We note that the partite sets of a bipartite graph are independent sets. The complete bipartite graph 𝐾𝑟 ,𝑠 is a bipartite graph with partite sets 𝑋 and 𝑌 , where |𝑋 | = 𝑟, |𝑌 | = 𝑠, and every vertex in 𝑋 is adjacent to every vertex in 𝑌 . The union 𝐺 = 𝐺 1 ∪ 𝐺 2 of two graphs 𝐺 1 and 𝐺 2 has vertex set 𝑉 (𝐺) = 𝑉 (𝐺 1 ) ∪ 𝑉 (𝐺 2 ) and edge set 𝐸 (𝐺 1 ) ∪ 𝐸 (𝐺 2 ). If 𝐺 is a union of 𝑘 copies of a graph 𝐹, we write 𝐺 = 𝑘 𝐹. For an integer 𝑘 ≥ 1, let [𝑘] = {1, 2, . . . , 𝑘 } and [𝑘] 0 = [𝑘] ∪ {0} = {0, 1, . . . , 𝑘 }. A walk in a graph 𝐺 from a vertex 𝑢 to a vertex 𝑣, called a (𝑢, 𝑣)-walk, is a finite alternating sequence of vertices and edges, starting with the vertex 𝑢 and ending with the vertex 𝑣, in which each edge of the sequence joins the vertex that precedes it in the sequence to the vertex that follows it in the sequence. A (𝑢, 𝑣)-trail is a (𝑢, 𝑣)-walk containing no repeated edges and a (𝑢, 𝑣)-path is a (𝑢, 𝑣)-walk

Chapter 1. In the Beginning: Roots of Domination in Graphs

4

containing no repeated vertices. A cycle is a closed (𝑢, 𝑣)-trail. The length of a path (respectively, cycle) equals the number of edges in the path (respectively, cycle). A graph of order 𝑛 which itself is a path is called the path 𝑃𝑛 . Thus, the path 𝑃𝑛 is the graph of order 𝑛 whose vertices can be labeled 𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 and whose edges are 𝑣 𝑖 𝑣 𝑖+1 for 𝑖 ∈ [𝑛 − 1]. For an integer 𝑛 ≥ 3, the cycle 𝐶𝑛 is the graph of order 𝑛 whose vertices can be labeled 𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 and whose edges are 𝑣 1 𝑣 𝑛 and 𝑣 𝑖 𝑣 𝑖+1 for 𝑖 ∈ [𝑛 − 1]. The cycle 𝐶𝑛 is also referred to as an 𝑛-cycle. We write 𝑃𝑛 : 𝑣 1 𝑣 2 . . . 𝑣 𝑛 and 𝐶𝑛 : 𝑣 1 𝑣 2 . . . 𝑣 𝑛 𝑣 1 to denote the paths and cycles, respectively, with vertex sequence (𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 ). Two vertices 𝑢 and 𝑣 are connected if there is a (𝑢, 𝑣)-path in 𝐺, and a graph 𝐺 is said to be connected if every two of vertices in 𝑉 are connected. The distance 𝑑 (𝑢, 𝑣) = 𝑑𝐺 (𝑢, 𝑣) between two vertices 𝑢 and 𝑣 in a connected graph 𝐺 is the minimum length of a (𝑢, 𝑣)-path in 𝐺. The eccentricity ecc(𝑣) = ecc𝐺 (𝑣) of a vertex 𝑣 in a connected graph 𝐺 is themaximum of the distances from 𝑣 to the other vertices of 𝐺; that is, ecc(𝑣) = max 𝑑 (𝑢, 𝑣) : 𝑢 ∈ 𝑉 . The diameter diam(𝐺) is the maximum eccentricity taken over all vertices of 𝐺 and the radius rad(𝐺) is the minimum eccentricity taken over all vertices of 𝐺. A vertex of 𝐺 with eccentricity equal to rad(𝐺) is called a central vertex. Abusing notation slightly, we refer to a central vertex as simply a center and say that a graph having exactly one central vertex 𝑥 is centered at 𝑥. A tree is an acyclic connected graph. A star is a tree with at most one vertex that is not a leaf, that is, a star is a tree with diameter at most 2. Thus, stars consist of complete bipartite graphs 𝐾1,𝑠 for 𝑠 ≥ 1 along with the trivial graph 𝐾1 . A double star 𝑆(𝑟, 𝑠), for 1 ≤ 𝑟 ≤ 𝑠, is a tree with exactly two (adjacent) vertices that are not leaves, with one of the vertices having 𝑟 leaf neighbors and the other 𝑠 leaf neighbors. The subdivision of edge 𝑢𝑣 ∈ 𝐸 consists of deleting the edge 𝑢𝑣 from 𝐸, adding a new vertex 𝑤 to 𝑉, and adding the new edges 𝑢𝑤 and 𝑤𝑣 to 𝐸. In this case, we say that the edge 𝑢𝑣 has been subdivided. In general, for an edge 𝑢𝑣 ∈ 𝐸 to be subdivided 𝑘 ≥ 1 times, we mean that edge 𝑢𝑣 is removed and replaced by a (𝑢, 𝑣)-path of length 𝑘 + 1. The subdivision graph 𝑆(𝐺) is the graph obtained from 𝐺 by subdividing every edge of 𝐺 exactly once.

1.3

Origins

In this section, we present the origins of domination in military tactics and chessboard problems.

1.3.1

Defensive and Offensive Strategies of the Roman Empire

In the fourth century AD, the Roman Empire dominated large areas of three continents, Europe, Africa, and Asia Minor. But it had begun to lose its power and it became increasingly difficult to secure all of its conquered regions. During the reign of Emperor Constantine the Great, who ruled between 306 and 337 AD, the Roman Empire controlled Britain, Gaul, Iberia (Spain and Portugal), southern Central Europe

Section 1.3. Origins

5

(including Italy), Asia Minor (including Turkey and Constantinople, a city named after the Emperor), and North Africa (including Egypt). Under Emperor Constantine, the Roman army was reorganized to consist of mobile field units and garrison soldiers, or local militia, capable of countering internal threats and barbarian invasions. A region was secured by armies being stationed there, and a region without an army was protected by sending mobile armies from neighboring regions. But Emperor Constantine decreed that a mobile field army could not be sent to defend a region if doing so left its original region unsecured. This defense strategy suggests a type of domination in graphs in which there are three types of vertices: unsecured (no armies), secured with one army (usually composed of local militia, which are not mobile armies), and secured with two armies (one being a highly trained, mobile army). The condition to be met is that every unsecured vertex must be adjacent to at least one vertex at which two armies are stationed. In this way, the set of vertices having one or two armies is a dominating set of the set of vertices having no armies. This defense strategy inspired the papers of Stewart [691] in 1999 and ReVelle and Rosing [658] in 2000, and then was formally defined as a type of domination in graphs for the first time in 2004 by Cockayne, Dreyer, Hedetniemi, and Hedetniemi [183].

1.3.2 Chaturanga Chaturanga is a war-oriented board game generally considered to have been developed in India during the sixth century AD. The name is a Sanskrit word meaning “four arms,” which stood for the four arms of the military, being the chariots, the cavalry, the elephants, and the infantry. Considered to be the precursor to the modern game of chess, chaturanga is a chesslike, two-player game played on a board of 8 × 8 squares, and with pieces very similar to those in chess: 1. Raja (king): moves one square in any direction. 2. Mantri (early form of queen): moves one square diagonally in any direction. 3. Ratha (rook): moves across any number of unoccupied squares either vertically or horizontally. 4. Gaja (elephant, early form of bishop): moves two squares diagonally but can jump over the first square. 5. Ashva (horse, knight): moves like the knight in chess, either two squares horizontally and then one square vertically, or two squares vertically and then one square horizontally, jumping over all intermediate squares. 6. Padáti (foot soldier, pawn): moves only one unoccupied square vertically, but can capture one square diagonally, as in chess. A capture in chaturanga consists of a piece of any type moving to a square, according to the rules for that piece, on which an opponent’s piece is found. The opponent’s piece is captured and removed from the game, and the piece that was moved to that square and made the capture remains on that square. In this way, every piece is said to dominate all squares it can reach in one move. Thus, the set of squares dominated by the pieces of one of the two players consists of all the squares occupied by the pieces plus all the squares which can be reached in one move by all

6

Chapter 1. In the Beginning: Roots of Domination in Graphs

of the pieces. Although there were other board games that preceded chaturanga, they are generally called race games in which the objective is to reach some designated location before your opponent. Chaturanga is one of the first games to consider the concept of capturing an opponent’s pieces, and hence the concept of domination first appears.

1.3.3 Eight Queens Problem A German chess player, named Max Bezzel [75], posed the following problem in the September 1848 issue of the chess journal Berliner Schachzeitung: Eight Queens Problem. In how many ways can 8 queens be placed on the squares of the 8 × 8 chessboard so that no two queens can attack each other, that is, no two queens lie on the same row, or the same column, or the same diagonal? A chess piece is said to cover (attack or dominate) any square on a chessboard that it can reach in a single move. For example, in one move a queen can move any number of unoccupied squares horizontally, vertically, or diagonally. Thus, a queen covers all of the squares in the same row, column, or diagonal as the square on which it is located, as illustrated in Figure 1.1. Figure 1.2 illustrates one placement of 8 pairwise non-attacking queens.

Z

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Z Z

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Z Z Z Z5XqZ Z Z Z Z Z Z Z Z Z Z

Figure 1.1 Moves of a queen on an 8 × 8 chessboard

The 8-Queens Problem quickly generalizes to the 𝑛-Queens Problem of placing 𝑛 queens on an 𝑛 × 𝑛 board so that no two queens attack each other. In graph theory terminology the 𝑛-Queens Problem is easily stated as that of finding a maximum independent set 𝑆 of 𝑛 vertices in the queens graph Q𝑛 . The queens graph Q𝑛 has a vertex set 𝑉 consisting of the 𝑛2 squares of an 𝑛 × 𝑛 chessboard, and two vertices are adjacent if and only if the corresponding squares lie on a common row, a common column, or a common diagonal. The vertex independence number 𝛼(Q𝑛 ) of the queens graph Q𝑛 , therefore, equals the maximum number of queens which can be placed on the 𝑛 × 𝑛 chessboard so that no two queens attack

Section 1.3. Origins

7

Z Z 5™Xq Z Z Z Z5XqZ Z 5X Z Z Z Zq ™Xq 5Z Z Z Z Z Z Z 5™Xq Z Z5XqZ Z Z Z Z5XqZ Z Z Z ™Xq Z Z 5 Figure 1.2 Maximum independent set of 8 queens

each other. It is obvious that 𝛼(Q𝑛 ) ≤ 𝑛, since any set of more than 𝑛 queens would have to contain two queens that lie on a common row, column, or diagonal. It remains to be shown that for any 𝑛, 𝛼(Q𝑛 ) = 𝑛. In 1910 Ahrens [9] was the first person to prove that for every positive integer 𝑛 ≥ 4, 𝛼(Q𝑛 ) = 𝑛, that is, one can always place 𝑛 queens on an 𝑛 × 𝑛 chessboard so that no two queens attack each other. The 8-Queens Problem, posed by Max Bezzel, was reported to have attracted the attention of the famous mathematician Gauss, but it was Dr. Franz Nauck [608, 609] who in 1850 pointed out, apparently without proof, that there were 92 different ways to place 8 non-attacking queens on the standard chessboard. These solutions fell into 12 classes, 11 of which yield 8 solutions by rotations and reflections, and the 12th solution generates another 4 solutions. In 1874 Pauls [630] was the first to prove that 92 is indeed the total number of solutions to the 8-Queens Problem. In 1892, although no proofs were given, W.W. Rouse Ball [51] correctly reported that for boards of sizes 4, 5, 6, 7, 8, 9, and 10, there are altogether 2, 10, 4, 40, 92, 342, and 724 solutions, respectively, to the 𝑛-Queens Problem.

1.3.4

Five Queens Problem Five Queens Problem. Show that 5 queens can be placed on the squares of the 8 × 8 chessboard so that every square is either occupied by a queen or is attacked by a queen. In how many ways can this be done?

It was known from the earliest times that five queens were sufficient to cover or dominate every square of the 8 × 8 chessboard; see for example Figure 1.3, in which the five queens mutually cover one another, and Figure 1.4, in which the five queens form an independent set. But this was quickly generalized to the following. Queens Domination Problem. What is the minimum number of queens which can be placed on an 𝑛 × 𝑛 chessboard so that every square is either occupied by a queen or is attacked by a queen?

8

Chapter 1. In the Beginning: Roots of Domination in Graphs

5™Xq Z Z Z Z Z Z Z 5™Xq Z Z Z Z 5™Xq Z Z 5™Xq Z Z Z Z Z Z Z Z ™Xq Z Z Z 5Z Z Z Z Z Z

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Figure 1.3 Five queens covering an 8 × 8 chessboard

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Z Z Z 5X Z Z Z Zq Z Z Z5XqZ 5™Xq Z Z Z Z Z5XqZ Z Z Z Z5XqZ Z Z Z Z Z Figure 1.4 Five independent queens covering an 8 × 8 chessboard

According to Gibbons and Webb [335], this problem was first stated by Abbe Durand in 1861, but was also given in 1862 by C.F. de Jaenisch [218], a Finnish and Russian chess player (1813–1872) and theorist, who in the 1840s was among the top chess players in the world. In graph theory terminology the Queens Domination Problem is to determine the queens domination number 𝛾(Q𝑛 ), that is, the minimum number of queens necessary to cover, or dominate, every square of an 𝑛 × 𝑛 chessboard. Although it proved to be relatively easy to determine the queens independence number 𝛼(Q𝑛 ) = 𝑛, after all these years since 1861, the determination of the value of 𝛾(Q𝑛 ) for all 𝑛 ≥ 1, remains an unsolved, and quite difficult, problem. In 1862 De Jaenisch [218] determined the queens domination number 𝛾(Q𝑛 ), for 𝑛 ∈ [8], to be 1, 1, 1, 2, 3, 3, 4, 5. In particular, he showed that 𝛾(Q8 ) = 5; see Figure 1.4. The values 𝛾(Q9 ) = 𝛾(Q10 ) = 𝛾(Q11 ) = 5 were correctly reported by Ahrens [9] in 1910; see for example Figure 1.5. These values have since been verified by computer programs.

Section 1.3. Origins

9

Z

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Z Z Z Z Z Z 5™Xq Z Z Z Z Z Z Z Z ™Xq Z Z Z Z 5Z Z Z Z Z Z Z Z Z 5™Xq Z Z Z Z Z Z Z Z Z Z Z Z Z 5™Xq Z Z Z Z Z Z Z Z Z 5™Xq Z Z Z Z Z Z Z Z Z Figure 1.5 Five queens covering an 11 × 11 chessboard

1.3.5

Queens Independent Domination Problem

The independent domination number 𝑖(Q𝑛 ) of the queens graph Q𝑛 was identified as an interesting problem by De Jaenisch [218], who in 1862 correctly gave the first eight values of 𝑖(Q𝑛 ), which are 1, 1, 1, 3, 3, 4, 4, 5; see Figure 1.4 for 𝑛 = 8. These have been verified by computer. It is interesting to note that 𝛾(Q5 ) = 3 < 𝑖(Q5 ) = 4 and 𝛾(Q6 ) = 3 < 𝑖(Q6 ) = 4, while 𝛾(Q7 ) = 4 = 𝑖(Q7 ) and 𝛾(Q8 ) = 5 = 𝑖(Q8 ). Determining the domination numbers and independent domination numbers of the queens graph seem to be extremely difficult problems. As noted in [446] and [626], only relatively few exact values of these two domination numbers of the queens graph are known. The value of 𝛾(Q𝑛 ) is either known, or known to be one of two consecutive values, for all 𝑛 ≤ 120 (see [626]). The known values of 𝛾(Q𝑛 ) and 𝑖(Q𝑛 ), for 4 ≤ 𝑛 ≤ 20, are summarized in Table 1.1; the values 𝛾(Q20 ) = 11 and 𝑖(Q19 ) = 𝑖(Q20 ) = 11 were discovered in the 2017 PhD thesis [77] of Bird at the University of Victoria; Bird [77] also found five other new values: 𝛾(Q22 ) = 12, 𝛾(Q24 ) = 13, 𝑖(Q22 ) = 12, 𝑖(Q23 ) = 13, and 𝑖(Q24 ) = 13. An independent covering of five queens for an 11 × 11 chessboard is illustrated in Figure 1.5.

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

𝛾(Q𝑛 ) 2 𝑖(Q𝑛 ) 3

3 3

3 4

4 4

5 5

5 5

𝑛

5 5

5 5

6 7

7 7

8 8

9 9

9 9

Table 1.1 First 20 values of 𝛾(Q𝑛 ) and 𝑖(Q𝑛 )

9 9 10 11 9 10 11 11

10

Chapter 1. In the Beginning: Roots of Domination in Graphs

1.3.6 Queens Total Domination Problem Still another interesting variant of the above three types of problems was formally introduced in 1892 by W.W. Rouse Ball [51]. Queens Total Domination Problem. What is the minimum number of queens which can be placed on an 𝑛 × 𝑛 chessboard so that every square is attacked by a queen, including the squares occupied by a queen? This is, of course, the total domination number 𝛾t (Q𝑛 ). Notice for example that Figure 1.3 shows five queens dominating the standard 8 × 8 chessboard, all of which lie on a common diagonal, and thus this set of five queens induces a connected subgraph, and thus this set is both a total dominating set and a connected dominating set. Hence, for 𝑛 = 8, 𝛾(Q8 ) = 𝛾t (Q8 ) = 5. At this point we have seen examples of dominating sets of queens of several different types, for example, dominating sets, maximum and minimum independent dominating sets, and total dominating sets. We next discuss these types of domination for different chess pieces.

1.3.7

Generalizations to Other Chess Pieces

Figure 1.6 W.W. Rouse Ball

In 1939 W.W. Rouse Ball [52] listed these three basic types of problems that were being studied on chessboards at the time. A photograph of Rouse Ball is given in Figure 1.6. • Covering: Determine the minimum number of chess pieces of a given type that are required to cover every square of an 𝑛 × 𝑛 chessboard (domination number). • Independent Covering: Determine the minimum number of mutually non-attacking chess pieces of a given type that are required to cover every square of an 𝑛 × 𝑛 chessboard (independent domination number).

Section 1.3. Origins

11

• Independence: Determine the maximum number of chess pieces of a given type that can be placed on an 𝑛 × 𝑛 chessboard such that no two pieces attack (cover) each other (independence number).

Z

Z

Z Z Z Z Z2UnZ –Un2UnZ2Un2–Un 2Z Z Z 2 –Un Z Z Z Z Z Z 2–Un Z Z 2–Un2UnZ2Un–Un 2Z Z Z Z2UnZ Z Z Z Z Z Z Figure 1.7 Twelve knights covering an 8 × 8 chessboard

2Un2–Un Z2UnZ Z Z Z Z Z2UnZ Z Z2Un2–Un Z Z 2 –Un Z Z Z Z Z Z Z 2–Un Z 2–Un2UnZ Z Z Z Zn 2U Z Z Z Z Z Z2Un2–Un Z2Un Figure 1.8 Fourteen independent knights covering an 8 × 8 chessboard

For example, Figure 1.7 shows a minimum set of 12 knights that dominate the 8×8 chessboard, while Figure 1.8 shows a minimum set of 14 knights that independently dominate the 8 × 8 chessboard. In fact, 𝛾(N8 ) = 12 < 𝑖(N8 ) = 14, where N8 is the knights graph defined as expected by the moves of a knight on an 8 × 8 chessboard. As another example, Figure 1.9 shows a minimum set of 8 bishops dominating the 8 × 8 chessboard. In 1954 and 1964, two Russian mathematicians and twin brothers, Akiva Moiseevich Yaglom (1921–2007) and Isaak Moiseevich Yaglom (1921–1988) produced a comprehensive collection of results about a wide variety of independent, dominating, and independent dominating solutions for a variety of chess pieces [758].

12

Chapter 1. In the Beginning: Roots of Domination in Graphs

Z

Z

Z

Z

Z

Z Z Z 4˜Wb4WbZ Z Z4WbZ Z Z Z Z4Wb Z Z Z 4˜Wb Z 4˜Wb4Wb4˜Wb Z Z Z Z Z Z Z Z Z Z Z Z Figure 1.9 Eight bishops covering an 8 × 8 chessboard

1.4

Application Driven Origins

In this section, we present a sample of applications that have prompted the growth of domination as a popular area of graph theory.

1.4.1

Radio Broadcasting

In his 1968 book, C.L. Liu [566] briefly discusses the notion of dominance as it applies to communication networks, in which a dominating set represents a set of designated nodes from which broadcast messages can be transmitted to all other nodes in the network. In his model, however, it was assumed that a broadcast station could only transmit messages to adjacent nodes. A more general graph theory model was presented by Erwin in 2004 [261] and subsequently by Dunbar, Erwin, Haynes, Hedetniemi, and Hedetniemi in 2006 [248], in which broadcast stations could be assigned varying amounts of broadcast power, which would enable them to transmit messages to nodes at greater distances. In order to explain this model we will need a few definitions. Recall that ecc(𝑣) denotes the eccentricity denotes the diameter of a graph 𝐺.  of a vertex 𝑣 and diam(𝐺) A function 𝑏 : 𝑉 → 0, 1, . . . , diam(𝐺) is called a broadcast if for every vertex 𝑣 ∈ 𝑉, 𝑏(𝑣) 𝑣 for which 𝑏(𝑣) > 0 are called broadcast vertices  ≤ ecc(𝑣). Vertices and 𝑉 + = 𝑣 : 𝑏(𝑣) > 0 . We say that a vertex 𝑣 hears a broadcast of a vertex 𝑤 ∈ 𝑉 +  if 𝑑 (𝑣, 𝑤) ≤ 𝑏(𝑤), and 𝐻 (𝑣) = 𝑤 ∈ 𝑉 + : 𝑑 (𝑣, 𝑤) ≤ 𝑏(𝑤) is the set of vertices which vertex 𝑣 can hear. of a broadcast vertex 𝑣 is defined as 𝑁 𝑏 [𝑣] =  The broadcast neighborhood 𝑤 : 𝑑 (𝑣, 𝑤) ≤ 𝑏(𝑣) , which is the set of vertices that can hear Í a broadcast from vertex 𝑣. The cost of a broadcast 𝑏 is defined to be 𝑏(𝑉) = 𝑣 ∈𝑉 + 𝑏(𝑣), the sum of the broadcast powers of all broadcast vertices. A broadcast 𝑏 is a dominating broadcast if 𝑁 𝑏 [𝑉 + ] = 𝑉, which means that every vertex hears at least one broadcast, that is, 𝐻 (𝑣) ≥ 1 for all 𝑣 ∈ 𝑉. Finally, the broadcast domination number 𝛾𝑏 (𝐺) of a graph 𝐺 equals the minimum cost of a dominating broadcast in 𝐺.

Section 1.4. Application Driven Origins

13

A dominating set 𝑆 in a graph 𝐺 = (𝑉, 𝐸) is said to be efficient if for every vertex 𝑣 ∈ 𝑉, |N[𝑣] ∩ 𝑆| = 1, that is, every vertex 𝑤 ∈ 𝑉 \ 𝑆 is adjacent to exactly one vertex in 𝑆 and no vertex in 𝑆 is adjacent to any other vertex in 𝑆. Not all graphs have efficient dominating sets, for example, the cycle 𝐶5 , and in general, most graphs do not have efficient dominating sets. However, for broadcast domination the situation is different. We say that a broadcast is efficient if every vertex hears exactly one broadcast, that is, if |𝐻 (𝑣)| = 1 for every 𝑣 ∈ 𝑉. Dunbar et al. [248] proved that every graph 𝐺 has an efficient dominating broadcast. Theorem 1.1 ([248]) The broadcast domination number 𝛾𝑏 (𝐺) of every connected graph 𝐺 can be achieved by an efficient dominating broadcast.

1.4.2

Computer Communication Networks

Most essential services for networked distributed systems, such as mobile or wired, are provided by maintaining a global predicate over the entire network, that is defined by some invariant components of the global state of the network. For example, a spanning tree of the network is maintained as each node of the network maintains links to its neighbors in the spanning tree in order to minimize latency and bandwidth requirements of multicast or broadcast messages, or to implement echo-based distributed algorithms [41]. In such a distributed computing environment, we are often faced with the problem of allocating a minimal number of a scarce or valuable hardware or software resources, such as high-performance graphics workstations, very large databases or file servers, to certain nodes in the network, in such a way that every node in the network not having such a resource has efficient access to each resource at a neighboring node. The nodes containing these resources, therefore, form a minimal dominating set. If a dominating set in a communication network represents a set of servers which provide an acceptable level of service and resources, then a total dominating set represents a similar set of servers with the added capability that each server is adjacent to at least one other server. Thus, each server has a backup, such that if it suffers a fault and its capability as a server is disabled, it can obtain backup from an adjacent server with a minimum delay. In this way, total dominating sets are more fault tolerant than dominating sets. Another useful set of nodes in a communication network is a connected dominating set 𝑆, which can serve as a communication backbone, because every node is either in the set 𝑆 or is adjacent to some node in 𝑆, and the fact that the nodes in 𝑆 form a connected subnetwork guarantees that any two nodes in 𝑆 can send messages to each other via a path of nodes in 𝑆. We say that a computer at a node 𝑢 can effectively access a resource at a node 𝑣 if 𝑑 (𝑢, 𝑣) ≤ 𝑑 for some suitably small distance 𝑑. If 𝑑 = 1, then the minimum number of copies of a given resource, which can be allocated to some of the nodes in a network so that all nodes either have the resource or have efficient access to a copy of the resource, is simply the domination number of the network. Suppose, furthermore, that we need to allocate many different resources, but all nodes have a fixed capacity 𝑟, which prevents us from assigning more than 𝑟 resources

14

Chapter 1. In the Beginning: Roots of Domination in Graphs

to any node. We wish to determine the maximum number 𝑅 (r,d) (𝐺) of different resources we can allocate to the nodes of a network 𝐺 such that (i) no more than 𝑟 different resources are allocated to any one node and (ii) every node has efficient 𝑑-access to every resource. An allocation that achieves this maximum is called an (𝑟, 𝑑)-configuration. For more information on (𝑟, 𝑑)-configurations, the reader is referred to the work of Fujita, Yamashita, and Kameda [314].

1.4.3

Sets of Representatives

Let the vertices of a graph represent the people in some organization. An edge between two people means that they know each other. We wish to form a committee with as few members as possible such that everyone not on the committee knows at least one member of the committee. Thus, we seek a minimum cardinality dominating set 𝑆 of this organization. The set 𝑆 could have another property of interest if every member of the committee knows at least one other member of the committee. In this case we would seek to find a minimum total dominating set of this organization. This brings to mind the following challenging problem from The World’s Hardest IQ Test by Scott Morris [600]. 51. One third of the members of a parliamentary body are elected every two years. The body has six committees. Each member of the body is a member of at least one committee, and no member is a member of more than two committees. No committee has more than eleven members. Each pair of committees has exactly two members in common. The Chairman is a member of the Rules Committee and of no other committee. Each member of the Budget Committee is also a member of another committee. The last digit of the number of members of the parliamentary body is: a: 2, b: 3, c: 4, d: 6, e: It cannot be determined from the information given. Next, let the vertices of a graph be of two kinds: (i) vertices representing the members of an organization and (ii) vertices representing areas of expertise. In this graph, an edge between a person 𝑢 and an area of expertise 𝑣 means that person 𝑢 has expertise in area 𝑣. In order to save costs, we want to select as few people as possible for a set 𝑆 such that for every area of expertise there is at least once person in 𝑆 who has expertise in this area. Thus, in this bipartite graph, we seek a minimum subset of the vertices/people that dominates all of the vertices/areas of expertise needed. One real world application of this type of domination occurs in making personnel assignments in the U.S. Navy. For example, the U.S. Navy has approximately 326,000 active personnel, many of whom periodically qualify for reassignments. In any given month, from 20,000 to 30,000 personnel are reassigned to fill a similar number of available positions requiring differing types of expertise.

Section 1.4. Application Driven Origins

15

1.4.4 School Bus Routing and Bus Stop Selection The School Bus Routing Problem, a complex and multi-faceted generalization of the well-known Vehicle Routing Problem, involves the routing, planning, and scheduling of public school bus transportation. The problem is typically divided into several subproblems, including (i) data preparation, which involves the determination of the road network, the location(s) of the school(s), locations of the homes of all students, and possible locations of bus stops, (ii) bus stop selection, in which bus stop locations are determined and all students are assigned to bus stops, and then (iii) determining the actual bus routes and their schedules, which must meet the constraints that for each student the distance travelled on the bus does not exceed the shortest distance from their home to the school by more than a given threshold. Subject to this constraint and bus capacity limit, the goal is to minimize the number of buses required. In bus stop selection, for schools in rural surroundings the students are assumed to be picked up at their homes. However, in urban areas the students are required to walk to a bus stop from their homes, this distance being no more than some specified maximum, and the actual walk must meet safety requirements, such as not crossing extremely busy highways. It must also be the case that a limit be placed on the number of students assigned to a given bus stop or series of consecutive bus stops, so as not to exceed the capacity of the bus. Considerations such as this give rise to the concept of capacitated domination as introduced in 2010 by Goddard, Hedetniemi, Huff, and McRae [345]. An 𝑟capacitated dominating set in a graph 𝐺 = (𝑉, 𝐸) is a set 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } for which there exists a partition {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } satisfying the following three conditions, for all 𝑖 ∈ [𝑘]: 1. 𝑣 𝑖 ∈ 𝑉𝑖 , 2. |𝑉𝑖 | ≤ 𝑟 + 1, and 3. 𝑣 𝑖 is adjacent to all vertices in 𝑉𝑖 \ {𝑣 𝑖 }. In an 𝑟-capacitated dominating set 𝑆, each vertex in 𝑆 dominates at most 𝑟 vertices in 𝑉 \ 𝑆, which corresponds to the maximum capacity of a school bus. The 𝑟-capacitated domination number 𝛾rc (𝐺) equals the minimum cardinality of an 𝑟-capacitated dominating set in 𝐺. In the special case that 𝑟 = 1, each vertex dominates at most one vertex in 𝑉 \ 𝑆, which means that the 1-capacitated domination number equals the minimum number of isolated vertices 𝐾1 and edges 𝐾2 into which the vertices can be partitioned; this equals what is known as the edge covering number 𝛽′ (𝐺). Two sources for information on school bus routing problem are by Park and Kim [629] and Bögl, Doerner, and Parragh [83].

1.4.5 Electrical Power Domination Electric power companies must continually monitor the state of their electric power networks. This includes such things as monitoring voltage magnitudes and machine phase angles at generators. One type of monitoring is to place phase measurement units, or PMUs, at selected locations in the network. Since PMUs are expensive, it is desirable to minimize their number, while still being able to monitor the entire network. A network is said to be completely observed if all of the state variables

16

Chapter 1. In the Beginning: Roots of Domination in Graphs

(e.g. voltages and currents) can be determined from the set of measurements being monitored. Let 𝐺 = (𝑉, 𝐸) be a graph representing an electric power network, where a vertex represents an electrical node (a substation bus where transmission lines, loads, and generators are connected), and an edge represents a transmission power line joining two electrical nodes. The problem of locating a smallest set of PMUs to monitor the entire network can be stated as follows. A PMU measures/observes the state variables (voltage and phase angle) at the vertex at which it is placed, along with the variables of all incident edges and the other vertices incident to these edges. These incident edges and neighboring vertices are also observed, according to the following rules: 1. Any vertex incident with an observed edge is observed. 2. Any edge between two observed vertices is observed. 3. If a vertex is incident to 𝑘 > 1 edges and if 𝑘 − 1 of these edges are observed, then all 𝑘 of these edges are observed. To illustrate this, let 𝑆 ⊂ 𝑉 be a set of vertices at which PMUs are located. Let 𝑆 = 𝑆0 ⊆ 𝑆1 ⊆ 𝑆2 ⊆ · · · be the sequence of successive sets defined as follows: (i) 𝑆1 = N[𝑆0 ], where N[𝑆0 ] equals the set of all vertices which are either in 𝑆0 or are adjacent to at least one vertex in 𝑆0 , (ii) for 𝑘 ≥ 2, 𝑆 𝑘 is obtained from 𝑆 𝑘−1 by adding to 𝑆 𝑘−1 all vertices 𝑤 ∈ 𝑉 \ 𝑆 𝑘−1 for which there exists a vertex 𝑣 ∈ 𝑆 𝑘−1 whose only neighbor in 𝑉 \ 𝑆 𝑘−1 is 𝑤. If there exists a 𝑘 such that 𝑆 𝑘 = 𝑉, then we say that 𝑆 is a power dominating set of 𝐺. The minimum cardinality of a power dominating set in 𝐺 is called the power domination number 𝛾 𝑃 (𝐺). This type of domination was introduced in 2002 by Haynes, Hedetniemi, Hedetniemi, and Henning [411].

1.4.6

Influence in Social Networks

The parameter which we introduce in this section is motivated by its applicability to the study of influence in social networks. Associated with each vertex 𝑣 ∈ 𝑉 in a social network, modeled by a graph 𝐺 = (𝑉, 𝐸) of order 𝑛, is an influence threshold 𝑡 (𝑣), where 𝑡 : 𝑉 → {0, 1, . . . , 𝑛 − 1} such that for every vertex 𝑣 ∈ 𝑉, 0 ≤ 𝑡 (𝑣) ≤ deg(𝑣). Let 𝑆 ⊂ 𝑉 be an arbitrary subset of 𝑉. We say that a vertex 𝑣 ∈ 𝑉 \ 𝑆 is influenced by the set 𝑆 if |N(𝑣) ∩ 𝑆| ≥ 𝑡 (𝑣), that is, 𝑣 has at least 𝑡 (𝑣) neighbors in 𝑆. The threshold 𝑡 (𝑣) is used as an indicator of the likelihood that 𝑣 will adopt a given product if a sufficient number 𝑡 (𝑣) of neighbors of 𝑣 also adopt a given product. The goal is to find a relatively small number of vertices in a network such that if the vertices in this set adopt a given product, then ultimately every vertex in the graph will also adopt the product. This influence threshold can be applied to many things, such as the likelihood that someone will vote for something if sufficiently many of their neighbors vote for it. It could also indicate the likelihood that someone will get a virus, if sufficiently many neighbors have the virus. In general, the influence value 𝑡 (𝑣) is used to predict that 𝑣 will make some decision if at least 𝑡 (𝑣) of the

Section 1.4. Application Driven Origins

17

neighbors of 𝑣 make the same decision. Note that the threshold 𝑡 (𝑣) can vary from vertex to vertex. For any sequence 𝜋 of vertices of a graph 𝐺, let 𝜋 𝑗 denote the set containing the first 𝑗 vertices in the sequence 𝜋. A set 𝑆 ⊆ 𝑉 is an influence set for a graph 𝐺 = (𝑉, 𝐸) of order 𝑛 = |𝑉 | if there exists a vertex sequence 𝜋 = (𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 ) such that for every 𝑗 ∈ [𝑛], either vertex 𝑣 𝑗 ∈ 𝑆 or 𝑣 𝑗 is influenced by 𝜋 𝑗 −1 . The minimum cardinality of an influence set for a graph 𝐺 with a threshold function 𝑡 is the influence domination number 𝛾(𝐺, 𝑡). In the early papers on this form of influence in social networks, the focus has been on studying the computational complexity of determining, or even approximating, this minimum number of initially selected vertices. The interested reader is referred to the 2014 paper by Bazgan, Chopin, Nichterlein, and Sikora [63].

1.4.7

Topographic Maps

In the field of surveying, a typical task is to produce a topographic map of a tract of land that records the positions and elevations of a carefully selected set of control points. A grid can be used in areas where the topography is fairly regular. The tract of land is divided uniformly into squares or rectangles by two sets of lines running in perpendicular directions and spaced uniformly apart. Once stakes are set at the intersections of these grid lines, it is then necessary to determine the elevations of all grid points. This is typically done with the use of a transit, which can measure horizontal and vertical distances between the transit and the stakes. From this information drafting instruments can automatically generate the contour lines. Since these are line-of-sight distance measurements, any obstruction, such as a building structure, a tree, a hill, a steep ravine or gully, can prevent a measurement of a grid point from being taken. In this case, the transit must be moved to another observation point from which no line-of-sight obstruction exists to the given grid point. Thus, surveyors seek to find a minimum number of control points, from at least one of which a line of sight measurement can be taken of any grid point. This is equivalent to finding a minimum, or at least minimal, dominating set of the line-of-sight graph for the given tract of land, the vertices of which correspond 1-1 with the grid points, and two vertices are adjacent if and only if the corresponding grid points permit a line-of-sight measurement.

1.4.8 Transporting Hazardous Materials The transportation of hazardous materials (hazmats) is an increasingly important activity, because of the steadily increasing number of shipments per day, more than one million a day in North America in 2007 [259], and the corresponding increase in the associated risks, including injuries, deaths, and millions of dollars in property damage resulting from incidents. Consequently, there are increasingly strict government regulations for the transportation of hazardous materials [637].

18

Chapter 1. In the Beginning: Roots of Domination in Graphs

Figure 1.10 Dénes Kőnig

One of the more important restrictions is that all such vehicles must be inspected at one or more intermediate locations while enroute to their destination, in order to ensure that the security of materials being transported is in continual compliance with regulations. This raises the problem of deciding where to locate inspection stations in a transportation network [69], which gives rise to this general problem. Assume that no hazardous materials shipment can travel a distance of 𝑘 without reaching at least one inspection station. A set 𝑆 ⊆ 𝑉 in a graph 𝐺 = (𝑉, 𝐸) is a 𝑘-path vertex cover if every path of order 𝑘 contains at least one vertex in 𝑆, or equivalently if no path 𝑃 𝑘 is contained in the subgraph 𝐺 [𝑉 \ 𝑆] induced by 𝑉 \ 𝑆. The 𝑘-path covering number 𝜓 𝑘 (𝐺) equals the minimum cardinality of a 𝑘-path vertex cover in 𝐺. By definition, if 𝑆 is a 𝑘-path vertex cover, then the subgraph 𝐺 [𝑉 \ 𝑆] does not contain a path of length 𝑘. Thus, if the vertices in 𝑆 are the inspection stations, then it is not possible for anyone to illegally transport hazardous materials over a distance of 𝑘 without being detected. The 𝑘-path covering number was introduced in 2011 by Brešar, Kardoš, Katrenič, and Semanišin [115].

1.5

Early Chronology of Domination in Graph Theory

As we have seen, the concept of domination has appeared under several different guises as early as the fourth century. Although domination in graphs was not formally defined in mathematics until the 1960s, the basic idea appeared in digraphs some thirty years earlier. In this section, we briefly review the sequence of books and papers in graph theory, which have served to provide the foundations of the theory of domination in graphs. We also give examples of some early results that shaped the field. These results will be presented in more detail in subsequent chapters.

Section 1.5. Early Chronology of Domination in Graph Theory

19

1936 Dénes Kőnig (1884–1944)

Although elements of graph theory can be found in a number of earlier sources, it is generally agreed that the first book written specifically about graph theory was by Dénes Kőnig [535] in 1936, entitled Theorie der Endlichen und Unendlichen Graphen, Kombinatorische Topologie der Streckenkomplexe. Kőnig was at that time professor of mathematics at the Royal Joseph University in Budapest (today known as Budapest University of Technology and Economics). See Figure 1.10 for a photograph of Kőnig. It is interesting to note that according to the Mathematics Genealogy Project, Kőnig’s PhD advisors, in 1907, were József Kürschák and Hermann Minkowski, while Kőnig himself had but one PhD student, the well-known Tibor Gallai (shown in Figure 1.14), who in turn had two PhD students, Jenö Lehel and László Lovász. In this book, Kőnig was perhaps the first person to formally define independent domination in digraphs. In brief, a digraph 𝐷 = (𝑉, 𝐴) consists of a set 𝑉 of vertices and a set 𝐴 of arcs or ordered pairs of vertices (𝑢, 𝑣), the arc being directed from vertex 𝑢 to vertex 𝑣, sometimes denoted by 𝑢 → 𝑣, in which case we say that vertex 𝑣 is an out-neighbor of vertex 𝑢 and vertex 𝑢 is an in-neighbor of vertex 𝑣. A directed path from a vertex 𝑢 to a vertex 𝑣 is a vertex sequence of the form 𝑢 = 𝑣 0 , 𝑣 1 , . . . , 𝑣 𝑘 = 𝑣 such that for every 𝑖 ∈ [𝑘], (𝑣 𝑖−1 , 𝑣 𝑖 ) is an arc in 𝐴. In [535] Kőnig defined a basis of a directed graph 𝐷 = (𝑉, 𝐴) to be a set 𝐵 ⊂ 𝑉 having the following two properties: (a) For every vertex 𝑣 ∈ 𝑉 \ 𝐵, there exists a vertex 𝑢 ∈ 𝐵 and a directed path from 𝑢 to 𝑣. (b) For every pair of vertices 𝑢, 𝑣 ∈ 𝐵, there is no directed path from 𝑢 to 𝑣. Notice that by (b), every basis of a directed graph 𝐷 is an independent set. Theorem 1.2 ([535]) Every finite directed graph 𝐷 = (𝑉, 𝐴) has a basis. Kőnig then defined a basis of the second kind to be a vertex set 𝐵 satisfying the following two conditions: (a) If 𝑣 is a vertex in 𝑉 \ 𝐵, then there is an arc (𝑢, 𝑣) from a vertex 𝑢 ∈ 𝐵 to 𝑣. (b) There is no arc between two vertices in 𝐵. In the case where a digraph 𝐷 is symmetric, meaning that whenever there is an arc (𝑢, 𝑣) ∈ 𝐴, then the arc (𝑣, 𝑢) is also in 𝐴, Kőnig’s basis of the second kind appears to be the first time in the literature where an independent dominating set is defined in an undirected graph. It also, of course, defines an independent dominating set in a digraph for the first time. Kőnig is also very well known for the following theorem. Theorem 1.3 ([535]) A graph 𝐺 = (𝑉, 𝐸) is bipartite if the vertex set 𝑉 can be partitioned into two independent sets, or equivalently, if 𝐺 contains no cycles of odd length. Recall that 𝛼′ (𝐺) and 𝛽(𝐺) denote the matching number and the vertex cover number of 𝐺, respectively. Theorem 1.4 ([535]) If 𝐺 is a bipartite graph, then 𝛼′ (𝐺) = 𝛽(𝐺).

20

Chapter 1. In the Beginning: Roots of Domination in Graphs

Figure 1.11 John von Neumann

Figure 1.12 Oskar Morgenstern

In terms of domination, a maximum matching 𝑀 is both a dominating set of edges, in that every edge not in 𝑀 must be adjacent to an edge in 𝑀, and an independent dominating set of edges. We should note that Theorem 1.4, proved in 1931 by Kőnig, was also independently proved in 1931 in the more general case of weighted graphs by Jenö Egerváry. Thus, this is often referred to as the Kőnig-Egerváry Theorem. 1939 Patrick Michael Grundy (1917–1958)

In 1939 Grundy [366] defined the following type of function 𝑔 : 𝑉 → {0, 1, . . . , 𝑛 − 1} on a digraph 𝐷 = (𝑉, 𝐴) of order 𝑛 = |𝑉 |. We say that such a function 𝑔 is a Grundy function if for every vertex 𝑢 ∈ 𝑉, 𝑔(𝑢) is the value not assigned 
smallest to an out-neighbor of 𝑢, that is, not in the set 𝑔 N(𝑢) = 𝑔(𝑣) : (𝑢, 𝑣) ∈ 𝐴 .  For any Grundy function 𝑔 it is easy to see that the set 𝑆 = 𝑢 : 𝑔(𝑢) = 0 is an independent dominating set, since no two vertices 𝑢 and 𝑣 that are adjacent can have 𝑔(𝑢) = 𝑔(𝑣) = 0, and any vertex 𝑤 ∈ 𝑉 \ 𝑆 must have at least one out-neighbor 𝑢 with 𝑤𝑢 ∈ 𝐴 and 𝑔(𝑢) = 0. 1944 John von Neumann (1903–1957) and Oskar Morgenstern (1902–1977)

In 1944 Hungarian-American mathematician, physicist, and computer scientist, John von Neumann and economist Oskar Morgenstern published their well-known book, Theory of Games and Economic Behavior [739], which founded the interdisciplinary field of research called game theory. They introduced the notion of a kernel into the theory of games under the name solution. We include photographs of von Neumann and Morgenstern in Figs. 1.11 and 1.12, respectively. A kernel in a digraph 𝐷 = (𝑉, 𝐴) is a set 𝑆 of vertices having the property that (i) for every vertex 𝑣 ∈ 𝑆, there is no arc (𝑣, 𝑤) to another vertex 𝑤 ∈ 𝑆, and (ii) for every vertex 𝑤 ∈ 𝑉 \ 𝑆, there is an arc (𝑤, 𝑣) from 𝑤 to some vertex 𝑣 ∈ 𝑆. In undirected graphs, kernels are nothing more than independent dominating sets, first discussed by Kőnig in 1936. Von Neumann and Morgenstern [739] described the following scenario. Suppose a set of 𝑛 players are given a set 𝑋 of situations to consider in terms of preference.

Section 1.5. Early Chronology of Domination in Graph Theory

21

Given two situations 𝑢 and 𝑣, if a subset of the players prefer situation 𝑢 to 𝑣 and can, by some means, make their point of view prevail, then we add an arc from 𝑣 to 𝑢, by which it is indicated that 𝑢 is preferred to 𝑣. In this way, the players construct a digraph 𝐷 = (𝑉, 𝐴) in which the out-neighbors 𝑢 of a vertex 𝑣, indicated by arcs (𝑣, 𝑢), are all the situations which have been decided to be more preferable than 𝑣. Independent dominating sets occur in the study of 2-person games, where they are called kernels. They are used to represent winning positions 𝑆 in a game, where an opponent makes a move starting from a winning position, but must always move to a non-winning position in 𝑉 \ 𝑆. Since the set 𝑆 is independent, the player cannot move from a winning position to another winning position. The player with a winning strategy, therefore always has a move from a non-winning position in 𝑉 \ 𝑆 to a winning position in 𝑆. Now, if a kernel can be found for this digraph 𝐷, then since 𝑆 is an independent set, no situation in 𝑆 is preferred to any other, and since 𝑆 is a dominating set, for any situation 𝑥 not in 𝑆, there must be a situation 𝑢 ∈ 𝑆 that is more preferable and hence there is an arc (𝑥, 𝑢) from 𝑥 to 𝑢. Research therefore focuses on the question: does a directed graph or undirected graph have a kernel? Subsequently many theorems emerged with either necessary or sufficient conditions under which a kernel will or will not exist. For example, from the previous subsection about Grundy functions, it is clear that if a digraph has a Grundy function, then it has a kernel. 1958 Claude Berge (1926–2002)

In 1958 French mathematician, Claude Berge, at Maître de Recherches au Centre National de la Recherche Scientifique, published the second book on graph theory, The Theory of Graphs and its Applications [67], after Kőnig’s book in 1936. Berge was the first to formally define and study the vertex independence number, which he called the coefficient of internal stability and the domination number, which he called the coefficient of external stability. Berge is pictured in Figure 1.13. Berge also defined a counterpart for domination in digraphs called the absorption number of a digraph 𝐷, to equal the minimum cardinality of an absorbant set, which is a vertex set 𝑆 having the property that for every vertex in 𝑣 ∈ 𝑉 \ 𝑆, there is an arc (𝑣, 𝑤) from 𝑣 to a vertex 𝑤 ∈ 𝑆, or equivalently, every vertex in 𝑉 \ 𝑆 is adjacent to a vertex in 𝑆. He gave as examples of minimum dominating sets the 5 minimum dominating queens in Figure 1.4, the 12 minimum dominating knights in Figure 1.7, and the 8 minimum dominating bishops in Figure 1.9. We will explore other results of Berge in subsequent chapters. Berge [67] was also the first to observe the following result. Theorem 1.5 ([67]) A set 𝑆 of vertices in a graph 𝐺 is an independent and dominating set of 𝐺 if and only if 𝑆 is maximal independent. 1959 Tibor Gallai (1912–1992)

In 1959 Hungarian mathematician Tibor Gallai (pictured in Figure 1.14), a student of Dénes Kőnig, proved the following classic theorem, where 𝛼(𝐺), 𝛽(𝐺), 𝛼′ (𝐺), and 𝛽′ (𝐺) denote the independence number, the vertex covering number, the matching number, and the edge covering number, respectively.

22

Chapter 1. In the Beginning: Roots of Domination in Graphs

Figure 1.13 Claude Berge

Figure 1.14 Tibor Gallai

Figure 1.15 Øystein Ore

Theorem 1.6 (Gallai’s Theorem [324]) If 𝐺 is a connected graph of order 𝑛 ≥ 2, then 𝛼(𝐺) + 𝛽(𝐺) = 𝛼′ (𝐺) + 𝛽′ (𝐺) = 𝑛. Since Gallai’s Theorem, many theorems of the general form A (𝐺) + B (𝐺) = 𝑛 have appeared in the literature, and are called Gallai theorems. See for example Cockayne, Hedetniemi, and Laskar [195]. 1962 Øystein Ore (1899–1968)

In 1962 Øystein Ore (see Figure 1.15), a Norwegian-American mathematician at Yale University, published what can be considered the third graph theory book, after those by Kőnig and Berge. In Chapter 13 Dominating Sets, Covering Sets, and Independent Sets, Ore [622] provides the first use of the word “domination,” for what Kőnig had previously referred to as a basis of the second kind and Berge had referred to as the coefficient of external stability. From Ore’s use forward the word ‘domination’ became the accepted terminology. Ore’s wording for a dominating set

Section 1.5. Early Chronology of Domination in Graph Theory

23

Figure 1.16 Vadim Vizing

applied to both directed and undirected graphs, as follows: “A subset 𝐷 of 𝑉 is a dominating set for 𝐺 when every vertex not in 𝐷 is the endpoint of some edge from a vertex in 𝐷.” Ore [622] provided the following well-known upper bound on the domination number. Theorem 1.7 (Ore’s Theorem [622]) If 𝐺 is an isolate-free graph of order 𝑛, then 𝛾(𝐺) ≤ 12 𝑛. 1963 Vadim Vizing (1937–2017)

In 1963 Ukrainian (former Soviet) graph theorist Vadim Vizing, pictured in Figure 1.16, raised a question about the domination number of the Cartesian product of two graphs 𝐺 and 𝐻, which has become the most famous conjecture in domination theory. We need to give the following definition. The Cartesian product of
two graphs 𝐺 = (𝑉, 𝐸) and 𝐻 = (𝑊, 𝐹) is the graph 𝐺 □ 𝐻 = 𝑉 × 𝑊, 𝐸 (𝐺 □ 𝐻) , where two vertices (𝑥 1 , 𝑦 1 ) and (𝑥2 , 𝑦 2 ) are adjacent in 𝐺 □ 𝐻 if and only if 𝑥 1 = 𝑥2 and 𝑦 1 𝑦 2 ∈ 𝐸 (𝐻), or 𝑥 1 𝑥2 ∈ 𝐸 (𝐺) and 𝑦 1 = 𝑦 2 . Conjecture 1.8 (Vizing’s Conjecture [732]) For every pair of graphs 𝐺 and 𝐻, 𝛾(𝐺 □ 𝐻) ≥ 𝛾(𝐺)𝛾(𝐻). Several hundred papers have now been written on Vizing’s Conjecture; the reader is referred to an entire chapter by Hartnell and Rall [396] in 1998, and a survey paper by Brešar, Dorbec, Goddard, Hartnell, Henning, Klavžar, and Rall [113] in 2012. We devote Chapter 18 of this book to this conjecture. 1974 Juho Nieminen

One of the earliest and most basic theorems about the domination number of a graph is the following, due to the Finnish mathematician Juho Nieminen (Finnish Academy, Helsinki). A pendant edge in a graph 𝐺 is any edge, one of whose vertices has degree one. Let 𝜀 𝑓 (𝐺) denote the maximum number of pendant edges in a spanning forest of 𝐺 (a forest is an acyclic graph). It can be seen that 𝜀 𝑓 (𝐺) equals

24

Chapter 1. In the Beginning: Roots of Domination in Graphs

the maximum number of edges in what is called a spanning star forest, that is, a disjoint collection of stars. In 1974 Nieminen [613] proved the following Gallai type theorem. Theorem 1.9 ([613]) If 𝐺 is a graph of order 𝑛, then 𝛾(𝐺) + 𝜀 𝑓 (𝐺) = 𝑛. 1975 Amram Meir and John Moon

In 1975 Meir and Moon (University of Alberta) [589] published a key paper relating packing and covering numbers of trees, as follows. Recall that 𝛼(𝐺) denotes the independence number and 𝜌(𝐺) denotes the packing number of a graph 𝐺. Theorem 1.10 ([589]) If 𝑇 is a tree of order 𝑛 ≥ 2, then 𝛼(𝑇) + 𝛾(𝑇) ≤ 𝑛. Corollary 1.11 ([589]) If 𝑇 is a tree of order 𝑛 ≥ 2, then 1 ≤ 𝛾(𝑇) ≤ 𝛼(𝑇) ≤ 𝑛 − 1.

𝑛 2

≤

We note that both Theorem 1.10 and its corollary generalize to all isolate-free graphs. Theorem 1.12 ([589]) For any nontrivial tree 𝑇, 𝜌(𝑇) = 𝛾(𝑇). Theorem 1.12 has become one of the best known results in domination theory. 1975 Ernest J. Cockayne, Seymour Goodman, and Stephen T. Hedetniemi

In 1975 Cockayne (University of Victoria) along with Goodman and Hedetniemi (University of Virginia) [187] published the first algorithm for computing the domination number of a tree, which executes in linear O (𝑛) time. 1977 E.J. Cockayne and S.T. Hedetniemi

In 1977 Cockayne and Hedetniemi (University of Oregon), published a paper, Towards a Theory of Domination in Graphs [194], which was one of the first to bring focus to domination in graphs as a field of study within graph theory, while citing the earlier work of Berge and Ore, mentioned previously. This paper, now having been cited more than 600 times, was noteworthy for introducing the notation 𝛾(𝐺) for the domination number and 𝑖(𝐺) for the independent domination number of a graph. The authors also introduced the concept of the domatic number dom(𝐺) and idomatic number idom(𝐺) of a graph, which are defined to equal the maximum orders of a partition of the vertices of a graph into dominating sets and independent dominating sets, respectively. There are by now nearly 250 papers on various aspects of domatic numbers in graphs. Domatic numbers and related partitions are discussed in Chapter 12. 1978 E.J. Cockayne, S.T. Hedetniemi, and Donald J. Miller

In 1978 Cockayne, Hedetniemi, and Miller (University of Victoria), published a paper, Properties of Hereditary Hypergraphs and Middle Graphs [196], which for the first time defined the irredundance numbers of a graph and presented what is called the Domination Chain of inequalities involving domination parameters, as follows.

Section 1.5. Early Chronology of Domination in Graph Theory

25

A set 𝑆 is called irredundant if for every vertex 𝑣 ∈ 𝑆, N[𝑣] \N[𝑆 \ {𝑣}] ≠ ∅. The lower and upper irredundance numbers ir(𝐺) and IR(𝐺) equal the minimum cardinality of a maximal irredundant set and the maximum cardinality of an irredundant set in 𝐺, respectively. The parameters ir(𝐺), IR(𝐺), and Γ(𝐺) were first defined in this paper. The domination number 𝛾(𝐺), the independent domination number 𝑖(𝐺), the independence number 𝛼(𝐺), and the upper domination number Γ(𝐺) have already been defined. The following inequalities are referred to as The Domination Chain. Theorem 1.13 ([196]) For any graph 𝐺, ir(𝐺) ≤ 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺) ≤ IR(𝐺). The authors also observed the following result, which is analogous to Berge’s Theorem 1.5. Theorem 1.14 ([196]) A set 𝑆 of vertices of a graph 𝐺 is an irredundant and dominating set of 𝐺 if and only if 𝑆 is minimal dominating. Indeed, it was later observed that if a set 𝑆 is a maximal independent set, then it is a minimal dominating set, and if a set 𝑆 is a minimal dominating set, then it is a maximal irredundant set. One final result in [196] is interesting to observe. The well-studied chromatic number 𝜒(𝐺) of a graph 𝐺 = (𝑉, 𝐸) equals the minimum order of a partition of 𝑉 into independent sets. One can define the independence graph 𝐼 (𝐺) of any graph 𝐺 to be the intersection graph of the set of all independent sets of vertices in 𝐺, that is, the vertices of 𝐼 (𝐺) correspond to the independent sets of vertices in 𝐺 and two vertices are adjacent in 𝐼 (𝐺) if and only if the corresponding independent sets have a vertex in common. Theorem 1.15 ([196]) For any graph 𝐺, 𝜒(𝐺) = 𝛾(𝐼 (𝐺)). 1980 E.J. Cockayne, Robyn Mason Dawes (1936–2010), and S.T. Hedetniemi

In 1980 Cockayne, Hedetniemi, and Dawes (Professor of Psychology at the University of Oregon) published a short note [182] which for the first time formally defined the total domination number of a graph, giving it the notation 𝛾t (𝐺). Clearly, many examples of total dominating sets of queens and other chess pieces had been observed, even some for more than 100 years, but the formal definition which applies to arbitrary graphs had not been defined. Today this paper has been cited more than 600 times, and more than 850 papers have been published on total domination alone. The reader is referred to the comprehensive 2013 book on total domination by Henning and Yeo [490]. 1998 Teresa W. Haynes, S.T. Hedetniemi, and Peter J. Slater (1946–2016)

In 1998 Haynes, Hedetniemi, and Slater published the first comprehensive treatment of domination in graphs in the two books: Fundamentals of Domination in Graphs [417] and Domination in Graphs: Advanced Topics [416]. The fundamentals book [417] has been cited more than 5000 times to date. Peter Slater is pictured

26

Chapter 1. In the Beginning: Roots of Domination in Graphs

in Figure 1.17. A list of other books containing information on domination can be found in Appendix B and a list of survey papers on domination can be found in Appendix C.

Figure 1.17 Peter J. Slater

Chapter 2

Fundamentals of Domination 2.1 Introduction As we have seen in Chapter 1, domination in graphs has roots in many sources, including defense strategies, games such as chess, computer communication networks, and network surveillance and security. In this chapter, we discuss the graph theoretical core concepts of domination and equivalent definitions for the domination number, thereby setting the foundation for the remaining chapters in the book. In order to explain the core concepts in domination in graphs, we need only a few definitions, which are given in the glossary in Appendix A and in Chapter 1.

2.2

Core Concepts

In this section, we discuss the core concepts of domination and develop what is called the Domination Chain, which was introduced in 1978 by Cockayne et al. [196]. The Domination Chain expresses relationships that exist among independent sets, dominating sets, and irredundant sets in graphs. This inequality sequence has become one of the major focal points in the study of domination in graphs, inspiring much interest and serving as a source for several hundred papers. Prior to stating the chain, we give a brief discussion of how this chain is developed using maximality and minimality conditions.

2.2.1 Independent Sets Definition 2.1 A set 𝑆 ⊆ 𝑉 of vertices in a graph 𝐺 is independent if no two vertices in 𝑆 are adjacent. The concept of domination in graphs can be said to originate in the definition of a maximal independent set, as follows. Let P denote any property of sets of vertices in a graph 𝐺 or a property of a graph 𝐺. A subset 𝑆 ⊆ 𝑉 having some property P is called a P-set, for example, © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_2

27

28

Chapter 2. Fundamentals of Domination

the property P0 of being an independent set, or the property PΔ𝑘 of being a set 𝑆 of vertices whose induced subgraph 𝐺 [𝑆] has maximum degree Δ(𝐺) ≤ 𝑘, for some positive integer 𝑘. A property P is called hereditary if every subset of a P-set is also a P-set, for example, the property P0 of being an independent set is hereditary, and so is property PΔ𝑘 . A property P is called superhereditary if every superset of a P-set is also a P-set, for example, the property of having a vertex that has at least 𝑘 neighbors in the set is superhereditary. A set 𝑆 is a maximal P-set if 𝑆 has property P but no proper superset 𝑆 ′′ , 𝑆 ⊂ 𝑆 ′′ , is a P-set. A set 𝑆 is a 1-maximal P-set if 𝑆 has property P but for every vertex 𝑤 ∈ 𝑉 \ 𝑆, the set 𝑆 ∪ {𝑤} does not have property P. A set 𝑆 is a minimal P-set if 𝑆 has property P but no proper subset 𝑆 ′ , 𝑆 ′ ⊂ 𝑆, is a P-set. A set 𝑆 is a 1-minimal P-set if 𝑆 has property P but for every vertex 𝑤 ∈ 𝑆, the set 𝑆 \ {𝑤} does not have property P. In 1997 Cockayne et al. [190] provided straightforward proofs of the following two results. Proposition 2.2 ([190]) For any graph 𝐺 and any superhereditary property P, a set 𝑆 ⊆ 𝑉 is a minimal P-set if and only if 𝑆 is a 1-minimal P-set. Proof By definition, every minimal P-set is a 1-minimal P-set. For the converse, let 𝑆 be a 1-minimal P-set, and let 𝑆 ′ be a proper subset of 𝑆. We wish to show that 𝑆 ′ is not a P-set, which would imply that 𝑆 is a minimal P-set. If |𝑆 ′ | = |𝑆| − 1, then 𝑆 ′ = 𝑆 \ {𝑣} for some vertex 𝑣 ∈ 𝑆, and the result follows from the definition of a 1-minimal P-set. Hence, we may assume |𝑆 ′ | ≤ |𝑆| − 2. Suppose, to the contrary, that 𝑆 ′ is a P-set. Let 𝑆 ′′ ⊂ 𝑆 be a superset of 𝑆 ′ such that |𝑆 ′′ | = |𝑆| − 1. We note that 𝑆 ′ ⊂ 𝑆 ′′ ⊂ 𝑆. Since P is a superhereditary property, the superset 𝑆 ′′ is a P-set. But this contradicts the supposition that 𝑆 is a 1-minimal P-set. Proposition 2.3 ([190]) For any graph 𝐺 and any hereditary property P, a set 𝑆 ⊆ 𝑉 is a maximal P-set if and only if 𝑆 is a 1-maximal P-set. Proof By definition, every maximal P-set is a 1-maximal P-set. For the converse, let 𝑆 be a 1-maximal P-set. If 𝑆 = 𝑉, then the result holds vacuously, so let 𝑆 ′′ be a proper superset of 𝑆. We wish to show that 𝑆 ′′ is not a P-set, which would imply that 𝑆 is a maximal P-set. If |𝑆 ′′ | = |𝑆| + 1, then the result follows from the definition of a 1-maximal P-set. Hence, we may assume |𝑆 ′′ | ≥ |𝑆| + 2. Suppose, to the contrary, that 𝑆 ′′ is a P-set. Let 𝑆 ′ be a subset of 𝑆 ′′ such that 𝑆 ⊂ 𝑆 ′ and |𝑆 ′ | = |𝑆| + 1. We note that 𝑆 ⊂ 𝑆 ′ ⊂ 𝑆 ′′ . Since P is a hereditary property, the subset 𝑆 ′ of 𝑆 ′′ is a P-set. But this contradicts the supposition that 𝑆 is a 1-maximal P-set. Since the property of being an independent set is hereditary, we can say by Proposition 2.3 that an independent set 𝑆 is maximal if and only if 𝑆 is a 1-maximal independent set. This is equivalent to saying that for every vertex 𝑣 ∈ 𝑉 \ 𝑆, the set 𝑆 ∪ {𝑣} is not an independent set. But this, in turn, is equivalent to the following property:

Section 2.2. Core Concepts

29

P ′ : for every vertex 𝑣 ∈ 𝑉 \ 𝑆, the induced subgraph 𝐺 [𝑆 ∪ {𝑣}] contains an edge between vertex 𝑣 and a vertex in 𝑆, which means that every vertex 𝑣 ∈ 𝑉 \ 𝑆 is adjacent to at least one vertex in 𝑆. Property P ′ then motivates the definition of dominating sets.

2.2.2 Dominating Sets Definition 2.4 A set 𝑆 ⊆ 𝑉 is a dominating set of a graph 𝐺 if every vertex 𝑣 ∈ 𝑉 \ 𝑆 is adjacent to at least one vertex in 𝑆. This, in turn, is equivalent to the following definition. Definition 2.5 A set 𝑆 ⊆ 𝑉 is a dominating set of a graph 𝐺 if N[𝑆] = 𝑉, that is, every vertex 𝑣 ∈ 𝑉 is either an element of 𝑆 or is in 𝑉 \ 𝑆 and is adjacent to at least one vertex in 𝑆. Definition 2.6 If 𝑋, 𝑌 ⊆ 𝑉, where the sets 𝑋 and 𝑌 are not necessarily disjoint, then 𝑋 dominates 𝑌 if 𝑌 ⊆ N[𝑋], that is, every vertex in 𝑌 belongs to 𝑋 or has a neighbor in 𝑋. In particular, if 𝑋 dominates 𝑉, then 𝑋 is a dominating set of 𝐺. Definition 2.7 The domination number 𝛾(𝐺) equals the minimum cardinality of a dominating set in 𝐺. The upper domination number Γ(𝐺) equals the maximum cardinality of a minimal dominating set in 𝐺. We say that any dominating set 𝑆 for which |𝑆| = 𝛾(𝐺) is a 𝛾-set of 𝐺, and any minimal dominating set 𝑆 for which |𝑆| = Γ(𝐺) is a Γ-set. Thus, the maximality condition for an independent set is identical to the condition that a set be a dominating set. And from this it follows immediately that every maximal independent set must also be a dominating set. Indeed, it was Berge [67, 68], as early as 1962, who first observed the following. Theorem 2.8 ([67, 68]) A subset of vertices in a graph 𝐺 is maximal independent if and only if it is independent and minimal dominating. Definition 2.9 The vertex independence number 𝛼(𝐺) equals the maximum cardinality of an independent set in 𝐺. The independent domination number 𝑖(𝐺) equals the minimum cardinality of a maximal independent set in 𝐺. An independent set of 𝐺 with cardinality 𝛼(𝐺) is called an 𝛼-set of 𝐺, while any maximal independent set of cardinality 𝑖(𝐺) is called an 𝑖-set of 𝐺. Thus, by definition, for any graph 𝐺, 𝑖(𝐺) ≤ 𝛼(𝐺). Notice that the property of being a dominating set 𝑆 is superhereditary, meaning that every proper superset of 𝑆 is also a dominating set. Thus, since every maximal independent set is also a minimal dominating set, we have the following inequalities, for any graph 𝐺: 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺). For example, the tree 𝑇 in Figure 2.1 has maximal independent sets of three cardinalities: {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 6 , 𝑣 7 }, {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 5 }, and {𝑣 4 , 𝑣 6 , 𝑣 7 }. Thus, 𝑖(𝑇) = 3 and

Chapter 2. Fundamentals of Domination

30 𝑣3

𝑣7

𝑣2 𝑣4

𝑣5 𝑣6

𝑣1 Figure 2.1 A tree 𝑇

𝛼(𝑇) = 5. This tree 𝑇 has exactly four minimal dominating sets: {𝑣 4 , 𝑣 5 }, {𝑣 4 , 𝑣 6 , 𝑣 7 }, {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 5 }, and {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 6 , 𝑣 7 }. Hence, 𝛾(𝑇) = 2 and Γ(𝑇) = 5. Since the property of being a dominating set is superhereditary, it follows from Proposition 2.2 that a set 𝑆 is a minimal dominating set if and only if it is a 1-minimal dominating set. Thus, for every vertex 𝑣 ∈ 𝑆, the set 𝑆 \ {𝑣} is not a dominating set. But this means that every minimal dominating set 𝑆 satisfies the following property:
P ′′ : for every vertex 𝑣 ∈ 𝑆, there exists a vertex 𝑤 ∈ 𝑉 \ 𝑆 \ {𝑣} which is not dominated by the set 𝑆 \ {𝑣}. This undominated vertex could be the vertex 𝑣 itself or a vertex 𝑤 ∈ 𝑉 \ 𝑆. In either case, it means that vertex 𝑣 dominates some vertex that no other vertex in 𝑆 dominates. This gives rise to the notion of a private neighbor with respect to a set of vertices, first introduced in 1978 by Cockayne et al. [196]. Definition 2.10 For a set 𝑆 ⊆ 𝑉 and a vertex 𝑣 ∈ 𝑆, the 𝑆-private neighborhood of 𝑣 is the set N[𝑣] \ N[𝑆 \ {𝑣}] and is denoted by pn[𝑣, 𝑆]. That is, pn[𝑣, 𝑆] equals the set of vertices that are in the closed neighborhood of 𝑣 but not in the closed neighborhood of the set 𝑆 \ {𝑣}. Equivalently, pn[𝑣, 𝑆] = 𝑤 ∈ 𝑉 : N[𝑤] ∩ 𝑆 = {𝑣} . If pn[𝑣, 𝑆] ≠ ∅, then we say that every vertex in pn[𝑣, 𝑆] is an 𝑆-private neighbor of 𝑣. Definition 2.11 For a set 𝑆 ⊆ 𝑉 and a vertex 𝑣 ∈ 𝑆, the open 𝑆-private neighborhood of 𝑣 is the set N(𝑣) \ N(𝑆 \ {𝑣}) and is denoted by pn(𝑣, 𝑆). That is, pn(𝑣, 𝑆) equals the set of vertices that are in the open neighborhoodof 𝑣 but not in the open neighborhood of the set 𝑆 \ {𝑣}. Equivalently, pn(𝑣, 𝑆) = 𝑤 ∈ 𝑉 : N(𝑤) ∩ 𝑆 = {𝑣} . If pn(𝑣, 𝑆) ≠ ∅, then we say that every vertex in pn(𝑣, 𝑆) is an open 𝑆-private neighbor of 𝑣. Definition 2.12 For a set 𝑆 ⊆ 𝑉 and a vertex 𝑣 ∈ 𝑆, the sets pn[𝑣, 𝑆] \ 𝑆 and pn(𝑣, 𝑆) \ 𝑆 are equal and we define the 𝑆-external private neighborhood of 𝑣 to be this set, abbreviated epn[𝑣, 𝑆] or epn(𝑣, 𝑆). Thus, epn[𝑣, 𝑆] = epn(𝑣, 𝑆). The 𝑆-internal private neighborhood of 𝑣 is defined by ipn[𝑣, 𝑆] = pn[𝑣, 𝑆] ∩ 𝑆 and its open 𝑆-internal private neighborhood is defined by ipn(𝑣, 𝑆) = pn(𝑣, 𝑆) ∩ 𝑆. We note that pn[𝑣, 𝑆] = ipn[𝑣, 𝑆] ∪ epn[𝑣, 𝑆] and pn(𝑣, 𝑆) = ipn(𝑣, 𝑆) ∪ epn(𝑣, 𝑆). To illustrate the private neighbor concept, we consider the tree 𝑇 in Figure 2.1. If 𝑆 = {𝑣 4 , 𝑣 5 }, then pn[𝑣 4 , 𝑆] = {𝑣 1 , 𝑣 2 , 𝑣 3 } since epn[𝑣 4 , 𝑆] = {𝑣 1 , 𝑣 2 , 𝑣 3 } and ipn[𝑣 4 , 𝑆] = ∅, and pn[𝑣 5 , 𝑆] = epn[𝑣 5 , 𝑆] = {𝑣 6 , 𝑣 7 }, while ipn[𝑣 5 , 𝑆] = ∅. Furthermore, pn(𝑣 4 , 𝑆) = {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 5 } since ipn(𝑣 4 , 𝑆) = {𝑣 5 } and epn(𝑣 4 , 𝑆) =

Section 2.2. Core Concepts

31

epn[𝑣 4 , 𝑆] = {𝑣 1 , 𝑣 2 , 𝑣 3 }. Similarly, pn(𝑣 5 , 𝑆) = {𝑣 4 , 𝑣 6 , 𝑣 7 } since ipn(𝑣 5 , 𝑆) = {𝑣 4 } and epn(𝑣 5 , 𝑆) = epn[𝑣 5 , 𝑆] = {𝑣 6 , 𝑣 7 }. If 𝑆 ′ = {𝑣 4 , 𝑣 6 , 𝑣 7 }, then pn[𝑣 4 , 𝑆 ′ ] = {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 } since epn[𝑣 4 , 𝑆 ′ ] = {𝑣 1 , 𝑣 2 , 𝑣 3 } and ipn[𝑣 4 , 𝑆 ′ ] = {𝑣 4 }. Also, pn[𝑣 6 , 𝑆 ′ ] = {𝑣 6 } and pn[𝑣 7 , 𝑆 ′ ] = {𝑣 7 } since ipn[𝑣 6 , 𝑆 ′ ] = {𝑣 6 }, ipn[𝑣 7 , 𝑆 ′ ] = {𝑣 7 }, and epn[𝑣 6 , 𝑆 ′ ] = epn[𝑣 7 , 𝑆 ′ ] = ∅. Furthermore, pn(𝑣 4 , 𝑆 ′ ) = {𝑣 1 , 𝑣 2 , 𝑣 3 } as ipn(𝑣 4 , 𝑆) = ∅ and epn(𝑣 4 , 𝑆 ′ ) = epn[𝑣 4 , 𝑆 ′ ] = {𝑣 1 , 𝑣 2 , 𝑣 3 }, and pn(𝑣 6 , 𝑆 ′ ) = ipn(𝑣 6 , 𝑆 ′ ) = epn(𝑣 6 , 𝑆 ′ ) = ∅. Similarly, pn(𝑣 7 , 𝑆 ′ ) = ipn(𝑣 7 , 𝑆 ′ ) = epn(𝑣 7 , 𝑆 ′ ) = ∅.

2.2.3

Irredundant Sets

The earlier observation that every vertex in a minimal dominating set dominates some vertex that no other vertex in 𝑆 dominates gives rise to the concept of irredundant sets. Irredundant sets were first defined by Cockayne et al. [196]. Definition 2.13 A set 𝑆 ⊆ 𝑉 is irredundant if for every vertex 𝑣 ∈ 𝑆, pn[𝑣, 𝑆] ≠ ∅. Definition 2.14 The irredundance number ir(𝐺) of a graph 𝐺 equals the minimum cardinality of a maximal irredundant set in 𝐺. The upper irredundance number IR(𝐺) equals the maximum cardinality of an irredundant set in 𝐺. For example, consider the graph 𝐺 in Figure 2.2. Each of the sets 𝑆 = {𝑣 1 , 𝑣 4 }, 𝑆 ′ = {𝑣 1 , 𝑣 5 }, and 𝑆 ′′ = {𝑣 1 , 𝑣 2 , 𝑣 3 } is a maximal irredundant set. Note that for 𝑆, pn[𝑣 1 , 𝑆] = {𝑣 2 , 𝑣 3 }, pn[𝑣 4 , 𝑆] = {𝑣 5 , 𝑣 6 }; for 𝑆 ′ , pn[𝑣 1 , 𝑆 ′ ] = {𝑣 1 , 𝑣 3 }, pn[𝑣 5 , 𝑆 ′ ] = {𝑣 5 , 𝑣 6 }; and for 𝑆 ′′ , pn[𝑣 1 , 𝑆 ′′ ] = {𝑣 4 }, pn[𝑣 2 , 𝑆 ′′ ] = {𝑣 5 }, and pn[𝑣 3 , 𝑆 ′′ ] = {𝑣 6 }. It can be shown that ir(𝐺) = 2 and IR(𝐺) = 3. 𝑣3

𝑣6 𝑣2

𝑣1

𝑣5 𝑣4

Figure 2.2 Graph 𝐺 with maximal irredundant sets of cardinality 2 and 3

We note that the domination number of the graph 𝐺 in Figure 2.2 is also 2, and so ir(𝐺) = 𝛾(𝐺) = 2. Slater provided one of the first known examples of a graph having irredundance number strictly less than its domination number. This graph, known as the Slater graph 𝐻, is illustrated in Figure 2.3. It can be shown that the set {𝑣 2 , 𝑣 3 , 𝑣 8 , 𝑣 9 } is a minimum maximal irredundant set for 𝐻 and the set {𝑣 2 , 𝑣 4 , 𝑣 6 , 𝑣 8 , 𝑣 10 } is a minimum dominating set of 𝐻, and so 4 = ir(𝐻) < 𝛾(𝐻) = 5. We note that attaching a leaf to a vertex of 𝐻 that is not in N[𝑣 6 ] creates another tree 𝑇 with ir(𝑇) = 4 < 5 = 𝛾(𝑇). It can easily be observed that the property of being an irredundant set is hereditary, namely, every subset of an irredundant set is also irredundant.

Chapter 2. Fundamentals of Domination

32 𝑣7

𝑣8

𝑣9

𝑣 10

𝑣 11

𝑣4

𝑣5

𝑣6 𝑣1

𝑣2

𝑣3

Figure 2.3 The Slater graph 𝐻

Proposition 2.15 ([196]) A subset of vertices in a graph 𝐺 is a minimal dominating set if and only if it is dominating and maximal irredundant. The following inequalities, called the Domination Chain, follow from the definitions, since every maximal independent set is a minimal dominating set and every minimal dominating set is a maximal irredundant set. Theorem 2.16 ([196]) For every graph 𝐺, ir(𝐺) ≤ 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺) ≤ IR(𝐺). As we have seen with the Slater graph, strict inequality between ir(𝑇) and 𝛾(𝑇) is possible for trees. On the other hand, in 1981 Cockayne et al. [185] proved that the three upper parameters of the Domination Chain are equal for bipartite graphs. Hence, the Domination Chain for bipartite graphs can be stated as follows. Theorem 2.17 ([185]) If 𝐺 is a bipartite graph, then ir(𝐺) ≤ 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) = Γ(𝐺) = IR(𝐺). Much research has focused on when equality is achieved between two parameters in the Domination Chain. In Chapter 5, we give a characterization of the trees having 𝛾(𝑇) = 𝑖(𝑇).

2.3

Parameters Suggested by the Definition of a Dominating Set

In this section, we define several types of dominating sets which naturally arise from different but equivalent definitions of a dominating set.

2.3.1

Total Dominating Sets

Perhaps the simplest and most natural variant of Definition 2.4, first formally defined in 1980 by Cockayne et al. [182], is the following, where all we do is change N[𝑆] = 𝑉 to N(𝑆) = 𝑉. Definition 2.18 A set 𝑆 ⊆ 𝑉 is a total dominating set, abbreviated TD-set, of a graph 𝐺 if N(𝑆) = 𝑉, that is, every vertex 𝑣 ∈ 𝑉 has at least one neighbor in 𝑆. The

Section 2.3. Parameters Suggested by the Definition of a Dominating Set

33

total domination number 𝛾t (𝐺) and the upper total domination number Γt (𝐺) equal the minimum and maximum cardinality, respectively, of a minimal TD-set in 𝐺. A TD-set 𝑆 for which |𝑆| = 𝛾t (𝐺) is a 𝛾t -set of 𝐺, and a TD-set 𝑆 for which |𝑆| = Γt (𝐺) is a Γt -set of 𝐺. Definition 2.19 If 𝑋, 𝑌 ⊆ 𝑉, where 𝑋 and 𝑌 are not necessarily disjoint, then the set 𝑋 totally dominates the set 𝑌 if every vertex in 𝑌 has a neighbor in 𝑋, that is, 𝑌 ⊆ N(𝑋). In particular, if 𝑋 totally dominates 𝑉, then 𝑋 is a TD-set of 𝐺. For example, the tree 𝑇 in Figure 2.1 has only one minimal TD-set, namely, the set {𝑣 4 , 𝑣 5 }, so 𝛾t (𝑇) = Γt (𝑇) = 2. Notice that if 𝑆 is a TD-set, then the induced subgraph 𝐺 [𝑆] is isolate-free. For that matter, the graph 𝐺 itself cannot have any isolated vertices, because every vertex in 𝑉 must have a neighbor in 𝑆. Thus, when speaking of the total domination number of a graph 𝐺, we must assume that the graph 𝐺 in question is isolate-free. The following observation is worth noting. Observation 2.20 For every isolate-free graph 𝐺, 𝛾(𝐺) ≤ 𝛾t (𝐺) ≤ Γt (𝐺). However, Γ(𝑇) and Γt (𝐺) are not comparable. For a star 𝐾1,𝑘 with 𝑘 ≥ 3, we have Γ(𝐾1,𝑘 ) = 𝑘 > Γt (𝐾1,𝑘 ) = 2, but for a complete graph 𝐾𝑛 , we have Γ(𝐾𝑛 ) = 1 < Γt (𝐾𝑛 ) = 2. Recall that a double star 𝑆(𝑟, 𝑠), for 1 ≤ 𝑟 ≤ 𝑠, is a tree with exactly two (adjacent) vertices that are not leaves, with one of the vertices having 𝑟 leaf neighbors and the other 𝑠 leaf neighbors. The double star 𝑆(2, 3) is illustrated in Figure 2.1 for example. If 𝐺 is a double star 𝑆(𝑟, 𝑠), then 𝛾(𝐺) = 𝛾t (𝐺) = 2, while Γ(𝐺) = 𝑟 + 𝑠 ≥ 2 = Γt (𝐺). The domination and total domination numbers of a path 𝑃𝑛 and cycle 𝐶𝑛 on 𝑛 ≥ 3 vertices are given by the following closed formulas, which will be discussed in more detail in Chapter 4. Observation 2.21 For 𝑛 ≥ 3, 𝛾(𝑃𝑛 ) = 𝛾(𝐶𝑛 ) =

𝑛 3

and 𝛾t (𝑃𝑛 ) = 𝛾t (𝐶𝑛 ) =

𝑛 2

+

𝑛 4

−

𝑛

.

4

By Observation 2.21, for example, we have 𝛾(𝑃12𝑘 ) = 4𝑘 < 6𝑘 = 𝛾t (𝑃12𝑘 ) for 𝑘 ≥ 1. For another example of strict inequality between the domination number and the total domination number, let 𝑇𝑘 be the graph formed from a star 𝐾1,𝑘 , for 𝑘 ≥ 2, by subdividing each edge exactly twice. For these trees 𝑇𝑘 , we have 𝛾(𝑇𝑘 ) = 𝑘 + 1 < 2𝑘 = 𝛾t (𝑇𝑘 ); see for example the tree 𝑇5 in Figure 2.4.

2.3.2

𝒌-Dominating Sets

Another definition equivalent to Definition 2.4 is the following.

Chapter 2. Fundamentals of Domination

34

Figure 2.4 𝛾(𝑇5 ) = 6 < 10 = 𝛾t (𝑇5 )

Definition 2.22 A set 𝑆 ⊆ 𝑉 is a dominating set of a graph 𝐺 if for every vertex 𝑣 ∈ 𝑉 \ 𝑆, |N(𝑣) ∩ 𝑆| ≥ 1. This definition suggests the following generalization, first introduced by Fink and Jacobson [296] in 1985. Definition 2.23 A set 𝑆 ⊆ 𝑉 is a 𝑘-dominating set of a graph 𝐺 if for every vertex 𝑣 ∈ 𝑉 \ 𝑆, |N(𝑣) ∩ 𝑆| ≥ 𝑘. The 𝑘-domination number 𝛾 𝑘 (𝐺) and upper 𝑘-domination number Γ𝑘 (𝐺) equal the minimum and maximum cardinality, respectively, of a minimal 𝑘-dominating set in 𝐺. The reader is referred to a chapter on 𝑘-domination by Haynes et al. [413].

2.3.3

𝑯-forming Dominating Sets

Another definition equivalent to Definition 2.4 is the following. Definition 2.24 A set 𝑆 ⊆ 𝑉 is a dominating set of a graph 𝐺 if every vertex 𝑣 ∈ 𝑉 \ 𝑆 is a vertex in a 𝐾2 -subgraph with at least one vertex in 𝑆. This definition suggests the following generalization, introduced in 2003 by Haynes et al. [415]. Definition 2.25 For a given graph 𝐻, a vertex set 𝑆 is an 𝐻-forming set if every vertex in 𝑉 \ 𝑆 is contained in a copy of 𝐻 with a subset of vertices in 𝑆. Equivalently, a set 𝑆 ⊆ 𝑉 is an 𝐻-forming set of 𝐺 if for every vertex 𝑣 ∈ 𝑉 \ 𝑆, there exists a subset 𝑅 ⊆ 𝑆, where |𝑅| = |𝑉 (𝐻)| − 1, such that 𝐺 [𝑅 ∪ {𝑣}] contains 𝐻 as a subgraph (not necessarily an induced subgraph). The minimum cardinality of a minimal 𝐻-forming set of 𝐺 is the 𝐻-forming number 𝛾 𝐻 (𝐺).

2.3.4

Perfect and Efficient Dominating Sets

The special case of Definition 2.22 when |N(𝑣) ∩ 𝑆| = 1 is worth consideration, as originally introduced by Bange et al. [56] in 1988, and independently by Dejter and Weichsel [220] in 1993 (see also [189] and [749]). Definition 2.26 A set 𝑆 ⊆ 𝑉 is called a perfect dominating set of a graph 𝐺 if for every vertex 𝑣 ∈ 𝑉 \ 𝑆, |N(𝑣) ∩ 𝑆| = 1, and is called an efficient dominating set if for every 𝑣 ∈ 𝑉, |N[𝑣] ∩ 𝑆| = 1.

Section 2.3. Parameters Suggested by the Definition of a Dominating Set

35

It should be pointed out, however, that not all graphs have an efficient dominating set, for example, the cycle 𝐶5 does not have an efficient dominating set, but any three consecutive vertices of 𝐶5 form a perfect dominating set. In addition, there are graphs whose only perfect dominating sets are the entire set 𝑉, such as the complete tripartite graph 𝐾2,2,2 or the smaller (in size) graph 𝐾 2 + 2𝐾2 , as given in [189]. We present more on perfect and efficient dominating sets in Chapter 9.

2.3.5

Distance-𝒌 Dominating Sets

Still another definition equivalent to Definition 2.4 is the following: Definition 2.27 A set 𝑆 ⊆ 𝑉 is a dominating set of a graph 𝐺 if for every vertex 𝑣 ∈ 𝑉 \ 𝑆, there exists a vertex 𝑤 ∈ 𝑆 such that 𝑑 (𝑣, 𝑤) = 1. This definition suggests the following generalization, first introduced in 1976 by Slater [678]. Definition 2.28 A set 𝑆 ⊆ 𝑉 is a distance-𝑘 dominating set of 𝐺 if for every vertex 𝑣 ∈ 𝑉 \ 𝑆, there exists a vertex 𝑤 ∈ 𝑆 such that 𝑑 (𝑣, 𝑤) ≤ 𝑘. The distance𝑘 domination number 𝛾 ≤ 𝑘 (𝐺) and upper distance-𝑘 domination number Γ≤ 𝑘 (𝐺) equal the minimum and maximum cardinality, respectively, of a minimal distance-𝑘 dominating set in 𝐺. The reader is referred to a chapter on distance domination by Henning in [413].

2.3.6

Fractional Domination

If 𝑆 ⊆ 𝑉 is a set of vertices of a graph 𝐺, we can define the characteristic function of 𝑆 to be the function 𝑓𝑆 : 𝑉 → {0, 1}, such that for every vertex 𝑣 ∈ 𝑉, 𝑓𝑆 (𝑣) = 1 if 𝑣 ∈ 𝑆, and 𝑓𝑆 (𝑣) = 0 if 𝑣 ∈ 𝑉 \ 𝑆. Given the characteristic function 𝑓𝑆 of a set 𝑆 ⊆ 𝑉, we can define
∑︁ 𝑓𝑆 N(𝑣) = 𝑓𝑆 (𝑤). 𝑤 ∈N(𝑣)

This gives rise to the next equivalent definition of a dominating set of a graph 𝐺 = (𝑉, 𝐸). Definition 2.29 A set 𝑆 ⊆ 𝑉 is a dominating set of 𝐺 if the characteristic function 𝑓𝑆
satisfies the condition that for every vertex 𝑣 ∈ 𝑉 \ 𝑆, 𝑓𝑆 N(𝑣) ≥ 1. Several variations are suggested by this definition. Definition 2.30 A set 𝑆 ⊆ 𝑉 is a dominating set of 𝐺 if the
characteristic function 𝑓𝑆 satisfies the condition that for every vertex 𝑣 ∈ 𝑉, 𝑓𝑆 N[𝑣] ≥ 1. Definition 2.31 A set 𝑆 ⊆ 𝑉 is a total dominating set of 𝐺 if the characteristic
function 𝑓𝑆 satisfies the condition that for every vertex 𝑣 ∈ 𝑉, 𝑓𝑆 N(𝑣) ≥ 1. The following definition of fractional domination was first given in 1987 by Hedetniemi et al. [444].

Chapter 2. Fundamentals of Domination

36

Definition 2.32 A function 𝑓 : 𝑉 → [0, 1], which assigns to each vertex 𝑣 ∈ 𝑉 a a fractional dominating rational number in the closed unit interval [0, 1] is called
N[𝑣] ≥ 1. The fractional dominafunction if for every vertex 𝑣 ∈ 𝑉, we have 𝑓 Í tion number 𝛾f (𝐺) equals the minimum 𝑣 ∈𝑉 𝑓 (𝑣) over all fractional dominating functions 𝑓 on 𝐺. The fractional version of total domination is defined as follows. Definition 2.33 A function 𝑓 : 𝑉 → [0, 1] is called a
fractional total dominating function if for every vertex 𝑣 ∈ 𝑉, we have 𝑓ÍN(𝑣) ≥ 1. The fractional total domination number 𝛾tf (𝐺) equals the minimum 𝑣 ∈𝑉 𝑓 (𝑣) over all fractional total dominating functions 𝑓 on 𝐺. The reader is referred to a chapter on fractional domination and fractional total domination by Goddard and Henning in [356]. We note that there are many other variations of domination that are not presented here. There is a discusison in [417] of parameters that arise from placing conditions on a dominating set and/or on the method of dominating. For example, if the subgraph 𝐺 [𝑆] induced by a dominating set 𝑆 is connected, then 𝑆 is a connected dominating set, while if 𝐺 [𝑆] has a perfect matching, then 𝑆 is a paired dominating set. Since our focus in this book is on domination, total domination, and independent domination, we refer the reader to the companion books [413, 414] for information on other types of domination in graphs.

2.4

Equivalent Formulations of Domination

In this section, we show that the concept of a dominating set in a graph has several equivalent formulations.

2.4.1 Pendant Edges in Spanning Forests A pendant edge in a graph 𝐺 = (𝑉, 𝐸) is any edge 𝑢𝑣 ∈ 𝐸 for which 𝑢 or 𝑣 is a leaf, that is, a vertex of degree 1. A graph 𝐺 is a forest if it is acyclic, that is, contains no cycles. Definition 2.34 Let 𝜀 𝑓 (𝐺) equal the maximum number of pendant edges in a spanning forest of 𝐺. In 1974 Nieminen [613] proved the following fundamental result relating the domination number of a graph 𝐺 to the maximum number of pendant edges in a spanning forest of 𝐺. Theorem 2.35 ([613]) For any graph 𝐺 of order 𝑛, 𝛾(𝐺) + 𝜀 𝑓 (𝐺) = 𝑛. Proof Let 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } ⊆ 𝑉 be any 𝛾-set of 𝐺. Construct a spanning forest 𝐹 of 𝐺 as follows. Begin by letting 𝐹 consist only of 𝑛 isolated vertices. For every vertex 𝑤 ∈ 𝑉 \ 𝑆, arbitrarily select an edge between 𝑤 and a vertex in 𝑆 and add

Section 2.4. Equivalent Formulations of Domination

37

it to 𝐹; since 𝑆 is a dominating set, every vertex in 𝑉 \ 𝑆 must be adjacent to at least one vertex in 𝑆, so there must be at least one such edge. The graph thus constructed is acyclic and has precisely |𝑉 \ 𝑆| edges, each of which is a pendant edge. Therefore, 𝜀 𝑓 (𝐺) ≥ |𝑉 \ 𝑆| = 𝑛 − 𝛾(𝐺). Conversely, let 𝐹 be a spanning forest of a graph 𝐺 containing 𝜀 𝑓 (𝐺) = 𝑘 pendant edges, 𝑒 1 , 𝑒 2 , . . . , 𝑒 𝑘 . For each pendant edge 𝑒 𝑖 , select a leaf 𝑤 𝑖 in 𝐹 incident to this edge, and let 𝑤 1 , 𝑤 2 , . . . , 𝑤 𝑘 be the set 𝑆 of 𝑘 selected leaves. It follows that the set 𝑉 \ 𝑆 is a dominating set of 𝐺, since every vertex in 𝑆 is a leaf in 𝐹 that is adjacent to a vertex in 𝑉 \ 𝑆. Thus, 𝛾(𝐺) ≤ |𝑉 \ 𝑆| = 𝑛 − 𝜀 𝑓 (𝐺), or 𝜀 𝑓 (𝐺) ≤ 𝑛 − 𝛾(𝐺).

2.4.2

Enclaveless Sets

We next define a type of set first introduced in 1977 by Slater [679]. Definition 2.36 For a graph 𝐺 = (𝑉, 𝐸) and a set 𝑆 ⊆ 𝑉, a vertex 𝑣 ∈ 𝑆 is called an enclave of 𝑆 if N[𝑣] ⊆ 𝑆. A set is said to be enclaveless if it does not contain any enclaves. In other words, every vertex in 𝑆 has a neighbor in 𝑉 \ 𝑆. The enclaveless number Ψ(𝐺) is the maximum cardinality of an enclaveless set of 𝐺 and 𝜓(𝐺) is the minimum cardinality of a maximal enclaveless set. Having the definition of an enclaveless set, one can prove the following, first shown in [679]. Theorem 2.37 ([679]) For any graph 𝐺 of order 𝑛, 𝛾(𝐺) + Ψ(𝐺) = 𝑛. Proof Let 𝐺 be a graph of order 𝑛, and let 𝑆 be a 𝛾-set in 𝐺. Consider the complement 𝑆 = 𝑉 \ 𝑆. It follows that 𝑆 is an enclaveless set, for if 𝑆 contains the closed neighborhood of some vertex 𝑣 ∈ 𝑆, then the vertex 𝑣 is not adjacent to any vertex in 𝑆, contradicting the assumption that 𝑆 is a dominating set. Since 𝑆 is an enclaveless set, it follows that Ψ(𝐺) ≥ |𝑆| = |𝑉 \ 𝑆| = 𝑛 − 𝛾(𝐺). Conversely, let 𝑆 ′ be a Ψ-set of 𝐺, that is, a maximum cardinality enclaveless set in 𝐺. Consider the set 𝑆 = 𝑉 \ 𝑆 ′ . Since 𝑆 ′ is enclaveless, every vertex in 𝑆 ′ has a neighbor in 𝑉 \ 𝑆 ′ = 𝑆. Thus, the set 𝑆 dominates the set 𝑆 ′ , and so 𝑆 is a dominating set of 𝐺. Therefore, 𝛾(𝐺) ≤ |𝑆| = |𝑉 \ 𝑆 ′ | = 𝑛 − Ψ(𝐺), or Ψ(𝐺) ≤ 𝑛 − 𝛾(𝐺). The simple proof of this theorem enables one to also prove the following two results. Corollary 2.38 For any graph 𝐺, the complement of any (minimal) dominating set is a (maximal) enclaveless set, and conversely, the complement of any (maximal) enclaveless set is a (minimal) dominating set. Corollary 2.39 For any graph 𝐺 of order 𝑛, Γ(𝐺) + 𝜓(𝐺) = 𝑛.

2.4.3

Spanning Star Partitions

Still another equivalent formulation of the domination number can be given, due to Hedetniemi [445] in 1983.

38

Chapter 2. Fundamentals of Domination

Definition 2.40 A spanning star partition of 𝐺 is a partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } of the vertex set of 𝐺 such that for all 𝑖 ∈ [𝑘], the induced subgraph 𝐺 [𝑉𝑖 ] contains a nontrivial spanning star, that is, 𝐺 [𝑉𝑖 ] has order 𝑛 ≥ 2 and contains a vertex which is adjacent to every other vertex in 𝐺 [𝑉𝑖 ]. Let 𝛼★ max (𝐺) denote the maximum order of a spanning star partition of 𝐺, and 𝛼★ min (𝐺) denote the minimum order of a spanning star partition of 𝐺. We note that the spanning star partition defined in Definition 2.40 is different than the definition of Acharya and Walikar [7], who in 1979 defined a star partition to be a vertex partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } such that for all 𝑖 ∈ [𝑘], the induced subgraph 𝐺 [𝑉𝑖 ] is a star, that is a graph of the form 𝐾1,𝑛𝑖 , where 𝑛𝑖 = 0 is allowed (𝐾1,0 is the trivial graph 𝐾1 ). Recall that 𝛼′ (𝐺) is the matching number of 𝐺, that is, the maximum cardinality of an independent set of edges in 𝐺. Theorem 2.41 ([445]) For any isolate-free graph 𝐺, 𝛼′ (𝐺) = 𝛼★ max (𝐺). Theorem 2.42 ([445]) For any isolate-free graph 𝐺, 𝛾(𝐺) = 𝛼★ min (𝐺). Proof Let 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } be a 𝛾-set of 𝐺. For each vertex 𝑣 𝑖 ∈ 𝑆, create a set 𝑉𝑖 containing vertex 𝑣 𝑖 , for 𝑖 ∈ [𝑘], as follows. For each vertex 𝑤 ∈ 𝑉 \ 𝑆, select one of its neighbors 𝑣 𝑗 ∈ 𝑆 (since 𝑆 is a dominating set, there must be at least one such neighbor) and place 𝑤 ∈ 𝑉 𝑗 . This will create a vertex partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } of 𝑉, such that each set 𝑉𝑖 contains a vertex 𝑣 𝑖 ∈ 𝑆 that is adjacent to every other vertex in the set. Among all such partitions, let 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } be one having a minimum number of subsets 𝑉𝑖 for which |𝑉𝑖 | = 1. If |𝑉𝑖 | ≥ 2 for each 𝑉𝑖 ∈ 𝜋, then 𝜋 defines a spanning star partition of 𝐺 having order 𝛾(𝐺). If not, renaming vertices in 𝑆 if necessary, assume without loss of generality that 𝑉1 = {𝑣 1 }. Since 𝐺 has no isolated vertices, 𝑣 1 must be adjacent to at least one other vertex 𝑤. If 𝑤 ∈ 𝑆, then the set 𝑆 \ {𝑣 1 } is a dominating set, contradicting the minimality of 𝑆. Hence, 𝑤 ∈ 𝑉 \ 𝑆. By the construction of the partition 𝜋, vertex 𝑤 must be in some 𝑉𝑖 of 𝜋 for 𝑖 ∈ [𝑘] \ {1}. Renaming vertices 𝑤 ∈ 𝑉2 , that is, both 𝑤 and 𝑣 2 are in the in 𝑆 if necessary, we may assume that
set 𝑉2 . If |𝑉2 | = 2, then 𝑆 \ {𝑣 1 , 𝑣 2 } ∪ {𝑤} is a dominating set of 𝐺, contradicting the minimality of 𝑆. Hence, |𝑉2 | ≥ 3, and  we can therefore form a new partition 𝜋 ′ = 𝑉1 ∪ {𝑤}, 𝑉2 \ {𝑤}, 𝑉3 , 𝑉4 , . . . , 𝑉𝑘 having fewer sets consisting of a single vertex, contradicting our choice of 𝜋. Hence, from an (arbitrary) 𝛾-set 𝑆, we can always construct a spanning star partition of order 𝛾(𝐺), and so 𝛼★ min (𝐺) ≤ 𝛾(𝐺). Conversely, let 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } be a spanning star partition of order 𝛼★ min (𝐺). The set 𝑆 consisting of the central vertices of the spanning stars in the induced subgraphs 𝐺 [𝑉𝑖 ] forms a dominating set. Thus, 𝛾(𝐺) ≤ 𝛼★ min (𝐺). We summarize the results of Theorems 2.35, 2.37, and 2.42 as follows. Corollary 2.43 If 𝐺 is a graph of order 𝑛, then 𝛾(𝐺) = 𝑛 − 𝜀 𝑓 (𝐺) = 𝑛 − Ψ(𝐺) = 𝛼★ min (𝐺).

Section 2.4. Equivalent Formulations of Domination

39

2.4.4 Non-dominating Partitions The domination and total domination numbers of any graph also have equivalent formulations in terms of what are called non-dominating partitions of a graph 𝐺 and its complement 𝐺, as introduced in 2016 by Desormeaux et al. [228]. Definition 2.44 A set of vertices that does not dominate 𝐺 is called a non-dominating set of 𝐺. In other words, for any non-dominating set 𝑆, there exists a vertex 𝑣 ∈ 𝑉 \ 𝑆 that has no neighbor in 𝑆, which means that 𝑣 is an enclave of 𝑉 \ 𝑆. A partition of the vertices of 𝐺 into non-dominating sets is a non-dominating partition. Let 𝜋nd (𝐺) equal the minimum order of a non-dominating partition of 𝐺. Definition 2.45 A set of vertices that does not totally dominate 𝐺 is called a nontotal dominating set of 𝐺, abbreviated non-TD-set. Equivalently, for any non-TD-set 𝑆, either 𝑆 is a non-dominating set or there is an isolated vertex in the subgraph 𝐺 [𝑆] induced by 𝑆. A partition of the vertices of 𝐺 into non-total dominating sets is a non-total dominating partition. Let 𝜋ntd (𝐺) equal the minimum order of a non-total dominating partition of 𝐺. If the domination number of a graph 𝐺 equals 1, then clearly, 𝐺 has a dominating vertex and hence, 𝐺 does not have a non-dominating partition. But the identity  partition, 𝜋 = {𝑣 1 }, {𝑣 2 }, . . . , {𝑣 𝑛 } , is a non-dominating partition for every graph 𝐺 with 𝛾(𝐺) ≥ 2. Also, since 𝑉 is a dominating set of 𝐺, any non-dominating partition must have a least two sets. Hence, we have the following observation. Observation 2.46 A graph 𝐺 of order 𝑛 has a non-dominating partition if and only if 𝐺 has no dominating vertex, in which case 2 ≤ 𝜋nd (𝐺) ≤ 𝑛. The next result relates the minimum order of a non-dominating partition of a graph 𝐺 to the total domination number of its complement 𝐺. Theorem 2.47 ([228]) If 𝐺 is a graph with no dominating vertex, then 𝛾t (𝐺) = 𝜋nd (𝐺). Proof Let 𝐺 be a graph with no dominating vertex, and let 𝜋 = {𝐴1 , 𝐴2 , . . . , 𝐴 𝑘 } be a non-dominating partition of 𝐺 of minimum order 𝜋nd (𝐺) = 𝑘. By Observation 2.46, we have 2 ≤ 𝜋nd (𝐺) ≤ 𝑛. We first show that 𝛾t (𝐺) ≤ 𝜋nd (𝐺). Since 𝜋 is a non-dominating partition of 𝐺, the set 𝐴𝑖 does not dominate 𝐺 for each 𝑖 ∈ [𝑘], implying that there exists a vertex 𝑎 𝑖 ∈ 𝑉 \ 𝐴𝑖 such that N(𝑎 𝑖 ) ∩ 𝐴𝑖 = ∅. Moreover, since 𝜋 has minimum cardinality, 𝑎 𝑖 is dominated by the set 𝐴 𝑗 for every 𝑗 ∈ [𝑘] \ {𝑖}; otherwise, the partition formed from 𝜋 by removing 𝐴𝑖 and 𝐴 𝑗 and adding 𝐴𝑖 ∪ 𝐴 𝑗 is a non-dominating partition of 𝐺 with order less than 𝜋nd (𝐺), a contradiction. Ð Hence, the non-dominated vertices 𝑎 𝑖 𝑘 are distinct for each 𝑖 ∈ [𝑘]. Note that 𝐴 = 𝑖=1 {𝑎 𝑖 } is a dominating set of 𝐺 of cardinality 𝑘, since 𝑎 𝑖 dominates every vertex in 𝐴𝑖 in the complement 𝐺. If 𝐴 is a TD-set of 𝐺, then 𝛾t (𝐺) ≤ 𝑘. If not, then it follows that there exists an 𝑎 𝑖 ∈ 𝐴 such that 𝑎 𝑖 is an isolated vertex of 𝐺 [ 𝐴]. By our selection of 𝑎 𝑖 , it follows that 𝑎 𝑖 ∉ 𝐴𝑖 . Hence, 𝑎 𝑖 ∈ 𝐴 𝑗 for some 𝑗 ∈ [𝑘] \ {𝑖}. Since the vertex 𝑎 𝑗 ∈ 𝐴 has no neighbor

40

Chapter 2. Fundamentals of Domination

in 𝐴 𝑗 in 𝐺, we note, in particular, that 𝑎 𝑗 is not adjacent to 𝑎 𝑖 in 𝐺. Hence, in 𝐺, the vertices 𝑎 𝑖 and 𝑎 𝑗 are adjacent, contradicting the fact that 𝑎 𝑖 is an isolate in 𝐺 [ 𝐴]. Thus, 𝐴 is a TD-set for 𝐺, and so 𝛾t (𝐺) ≤ | 𝐴| = 𝑘 = 𝜋nd (𝐺). Since 𝐺 has no dominating vertex, 𝐺 has no isolated vertex, that is, the total domination number of 𝐺 is defined. To see that 𝛾t (𝐺) ≥ 𝜋nd (𝐺), let 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 ℓ } be a 𝛾t -set of 𝐺, and so ℓ = 𝛾t (𝐺). For 𝑖 ∈ [ℓ], let 𝐵𝑖 = N𝐺 (𝑣 𝑖 ). Since 𝑆 is a TD-set of 𝐺, every vertex of 𝑉 belongs to some 𝐵𝑖 . Moreover, since 𝑆 is a 𝛾t -set of 𝐺, we have pn𝐺 (𝑣 𝑖 , 𝑆) ≠ ∅ and pn𝐺 (𝑣 𝑖 , 𝑆) ⊆ 𝐵𝑖 for each 𝑖 ∈ [ℓ]. We partition the vertices of 𝑉 as follows: let 𝐵1′ = 𝐵1 . For each 𝑗 ≥ 2, form 𝐵′𝑗 by removing the vertices Ð 𝑗 −1 from 𝐵 𝑗 that are contained in 𝑖=1 𝐵𝑖 . Note that pn𝐺 (𝑣 𝑖 , 𝑆) ⊆ 𝐵𝑖′ , and so 𝐵𝑖′ ≠ ∅ for 𝑖 ∈ [ℓ]. Note also that the vertex 𝑣 𝑖 ∉ 𝐵𝑖′ and 𝑣 𝑖 is not dominated by 𝐵𝑖′ in 𝐺 for 𝑖 ∈ [ℓ]. Hence, each 𝐵𝑖′ is a non-dominating set of 𝐺. Thus, 𝜋 = {𝐵1′ , 𝐵2′ , . . . , 𝐵ℓ′ } is a partition of 𝑉 into non-dominating sets of 𝐺, implying that 𝜋nd (𝐺) ≤ |𝜋| = ℓ = 𝛾t (𝐺). Consequently, 𝛾t (𝐺) = 𝜋nd (𝐺). A proof similar to the one used to prove Theorem 2.47 yields the following result for the domination number. Theorem 2.48 ([228]) For any graph 𝐺, 𝛾(𝐺) = 𝜋ntd (𝐺).

2.4.5

Total Domination and Splitting Graphs

Still another equivalent formulation of the total domination number in terms of the domination number can be given as follows. Definition 2.49 The splitting graph 𝐺 2 = (𝑉 ∪ 𝑉 ′ , 𝐸 ∪ 𝐸 ′ ) of an isolate-free graph 𝐺 = (𝑉, 𝐸) is the graph obtained from 𝐺 by adding to it a set 𝑉 ′ of vertices consisting of one copy 𝑣 ′ of every vertex 𝑣 ∈ 𝑉, and adding a set 𝐸 ′ of edges of the form 𝑣 ′ 𝑤, where 𝑣 ′ ∈ 𝑉 ′ , 𝑤 ∈ 𝑉, and 𝑣𝑤 ∈ 𝐸. Thus, the copy 𝑣 ′ ∈ 𝑉 ′ of the vertex 𝑣 ∈ 𝑉 is joined in 𝐺 2 to all neighbors of 𝑣 in 𝐺, that is, N𝐺2 (𝑣 ′ ) = N𝐺 (𝑣). We note that the set 𝑉 ′ is an independent set. Theorem 2.50 For any isolate-free graph 𝐺, 𝛾(𝐺 2 ) = 𝛾t (𝐺). Proof Let 𝑆 be a 𝛾t -set of 𝐺. Since 𝑆 is a TD-set of 𝐺, every vertex in 𝑉 has a neighbor in the set 𝑆 in 𝐺. Thus, the set 𝑆 totally dominates the set 𝑉 in 𝐺, and therefore the set 𝑉 in 𝐺 2 (noting that 𝐺 2 [𝑉] = 𝐺). Adopting the notation in Definition 2.49, let 𝑣 ′ ∈ 𝑉 ′ be the copy of the vertex 𝑣 ∈ 𝑉. Let 𝑥 be a neighbor of 𝑣 in 𝐺 that belongs to the TD-set 𝑆 of 𝐺. By construction, the vertices 𝑥 and 𝑣 ′ are adjacent in 𝐺 2 , implying that the set 𝑆 totally dominates the set 𝑉 ′ in 𝐺 2 . Hence, the set 𝑆 totally dominates the set 𝑉 ∪ 𝑉 ′ in 𝐺 2 , that is, 𝑆 is a TD-set of 𝐺 2 , and so 𝛾(𝐺 2 ) ≤ 𝛾t (𝐺 2 ) ≤ |𝑆| = 𝛾t (𝐺). Conversely, we must show that 𝛾t (𝐺) ≤ 𝛾(𝐺 2 ). Among all 𝛾-sets of 𝐺 2 , let 𝑆 have a minimum number of vertices in 𝑉 ′ . We show firstly that 𝑆 ⊆ 𝑉, that is, 𝑆 has ′ ′ ′ no vertices in 𝑉 ′ . Suppose, to the contrary,
that 𝑆 contains a vertex 𝑣 ∈ 𝑉 . If 𝑣 has ′ a neighbor 𝑥 ∈ 𝑆, then the set 𝑆 \ {𝑣 } ∪ {𝑣} is also a 𝛾-set of 𝐺 2 having fewer

Section 2.4. Equivalent Formulations of Domination

41

vertices in 𝑉 ′ than 𝑆, contradicting the choice of 𝑆. Hence, 𝑣 ′ has no neighbor in 𝑆, implying that the vertex 𝑣 has no neighbor in 𝑆. Thus, 𝑣 must be in 𝑆 and both 𝑣 ′ and 𝑣 are isolated vertices in 𝐺 [𝑆]. Now since 𝐺 is isolate-free, the vertices 𝑣 ′ and
𝑣 must have a common neighbor 𝑥 ∉ 𝑆. We note that 𝑥 ∈ 𝑉. It follows that 𝑆 \ {𝑣 ′ } ∪ {𝑥} is a 𝛾-set of 𝐺 2 having fewer vertices in 𝑉 ′ than 𝑆, a contradiction. Therefore, 𝑆 ⊆ 𝑉. We show next that there are no isolated vertices in 𝐺 [𝑆]. Suppose, to the contrary, that there exists an isolated vertex 𝑣 in 𝐺 [𝑆]. Since no neighbor of 𝑣 is in 𝑆, then no neighbor of 𝑣 ′ is in 𝑆 either. Thus, either 𝑣 ′ ∈ 𝑆, which contradicts our earlier observation that 𝑆 ⊆ 𝑉, or 𝑣 ′ ∉ 𝑆, which contradicts the fact that 𝑆 is a dominating set of 𝐺 2 . Hence, there are no isolated vertices in 𝐺 [𝑆], implying that 𝑆 is a TD-set of 𝐺, and hence 𝛾t (𝐺) ≤ |𝑆| = 𝛾(𝐺 2 ).

2.4.6

Dominating Sets and (1, 4 : 3)-Sets

Definition 2.51 A set 𝑆 ⊆ 𝑉 is called an (𝑖, 𝑗)-set in a graph 𝐺 if every vertex in 𝑉 \ 𝑆 is at most distance 𝑖 from at least one vertex in 𝑆 and is at most distance 𝑗 from a second, distinct vertex in 𝑆. The (𝑖, 𝑗)-domination number 𝛾 (𝑖, 𝑗 ) (𝐺) equals the minimum cardinality of an (𝑖, 𝑗)-set in 𝐺. Notice that by definition every (1, 𝑗)-set is a dominating set, and therefore, for any nontrivial, connected graph 𝐺, we have 𝛾(𝐺) ≤ 𝛾 (1, 𝑗 ) (𝐺). Definition 2.52 A set 𝑆 ⊆ 𝑉 is an (𝑖, 𝑗 : 𝑘)-set of 𝐺 if 𝑆 is an (𝑖, 𝑗)-set and every vertex in 𝑆 is at most distance 𝑘 from a second vertex in 𝑆. The next theorem was proved in 2008 by Hedetniemi et al. [443]. Theorem 2.53 ([443]) Every nontrivial dominating set of a connected graph 𝐺 is a (1, 4 : 3)-set. Proof Let 𝑆 ⊆ 𝑉 be any dominating set in a connected graph 𝐺, where we assume that |𝑆| ≥ 2. We will first show that for any vertex 𝑢 ∈ 𝑆, there exists another vertex 𝑤 ∈ 𝑆 such that 𝑑 (𝑢, 𝑤) ≤ 3. Suppose, to the contrary, that there exists a vertex 𝑢 ∈ 𝑆 such that the minimum distance between 𝑢 and another vertex in 𝑆 \ {𝑢} is at least 4. Let 𝑢, 𝑥, 𝑦 be the first three vertices on a shortest path from vertex 𝑢 to some vertex 𝑤 ∈ 𝑆, where necessarily 𝑥, 𝑦 ∈ 𝑉 \ 𝑆 and 𝑑 (𝑢, 𝑤) > 3. Notice that vertex 𝑢 cannot be adjacent to vertex 𝑦, else this is not a shortest path from 𝑢 to 𝑤. But since 𝑆 is a dominating set, vertex 𝑦 must be adjacent to at least one vertex 𝑧 ∈ 𝑆. It follows therefore that 𝑑 (𝑢, 𝑧) ≤ 3, contradicting our assumption. Thus, 𝑑 (𝑢, 𝑤) ≤ 3 for some 𝑤 ∈ 𝑆. Since 𝑆 is a dominating set, every vertex in 𝑣 ∈ 𝑉 \ 𝑆 is adjacent to some vertex in 𝑆. Further, since every vertex in 𝑆 is at most distance 3 to some other vertex in 𝑆, 𝑣 is at most distance 4 to a second vertex in 𝑆. It follows that the set 𝑆 is (1, 4 : 3)-set. It is important to observe that the (1, 4 : 3) bounds in Theorem 2.53 can be achieved, and thus the values of 𝑗 = 4 and 𝑘 = 3 cannot be decreased for arbitrary dominating sets in connected graphs. The set 𝑆 = {𝑣 2 , 𝑣 5 } in the path 𝑃6 in Figure 2.5 can be seen to be a (1, 4 : 3)-set.

Chapter 2. Fundamentals of Domination

42

Corollary 2.54 For any connected graph 𝐺, 𝛾(𝐺) = 𝛾 (1,4) (𝐺).

𝑣1

𝑣2

𝑣3

𝑣4

𝑣5

𝑣6

Figure 2.5 A (1, 4 : 3)-set of 𝑃6 Notice that the path 𝑃6 has only one 𝛾-set, namely the set 𝑆 = {𝑣 2 , 𝑣 5 }, and since this set 𝑆 is a (1, 4)-set, we can conclude that 𝛾(𝑃6 ) = 2 = 𝛾 (1,4) (𝑃6 ). We can also conclude that the path 𝑃6 does not have a (1, 3)-set of cardinality 2. However, the set 𝑆 ′ = {𝑣 1 , 𝑣 4 , 𝑣 6 } is both a (1, 3)-set and a (1, 2)-set, and the set 𝑆 ′′ = {𝑣 1 , 𝑣 3 , 𝑣 5 , 𝑣 6 } is a smallest (1, 1)-set. Notice that a (1, 1)-set is, in fact, a 2-dominating set in that every vertex in 𝑉 \ 𝑆 has at least two neighbors in 𝑆. Notice also that a (1, 4)-set is also a (1, 𝑗)-set for every 𝑗 ≥ 5. Thus, when we say that every dominating set in a graph 𝐺 is a (1, 4)-set, it might also be a (1, 3)-set, a (1, 2)-set, or a (1, 1)-set as well; but every (1, 3)-set, (1, 2)-set, and (1, 1)-set is always a (1, 4)-set. In 2012 Hedetniemi et al. [438] proved the following: (i) Every independent dominating set is a (1, 4 : 3)-set. (ii) Every total dominating set is a (1, 2 : 1)-set. (iii) For every graph 𝐺, 𝛾(𝐺) = 𝛾 (1,4) (𝐺) ≤ 𝛾 (1,3) (𝐺) ≤ 𝛾 (1,2) (𝐺) ≤ 𝛾 (1,1) (𝐺) = 𝛾2 (𝐺).

2.5

Domination in Terms of Perfection in Graphs

To the definitions of independent sets, dominating sets, and irredundant sets, we now introduce several concepts and parameters having to do with what is called perfection in graphs. These concepts were first introduced by Fricke et al. [310] in 1999 and later developed in detail by Hedetniemi et al. [439] in 2013. In the remainder of this section, we review the results given in [439]. Definition 2.55 Given a set 𝑆 ⊆ 𝑉 in a graph 𝐺, a vertex 𝑣 ∈ 𝑉 is said to be 𝑆-perfect if |N[𝑣] ∪ 𝑆| = 1, that is, the closed neighborhood N[𝑣] contains exactly one vertex in 𝑆. Definition 2.56 Given a set 𝑆 ⊆ 𝑉 in a graph 𝐺, a vertex 𝑣 is almost 𝑆-perfect if it is either 𝑆-perfect or is adjacent to an 𝑆-perfect vertex. When a set 𝑆 has been given and is assumed, we simply say that a vertex is perfect or almost perfect without referring to the set 𝑆. Definition 2.57 A set 𝑆 ⊆ 𝑉 is internally perfect if every vertex 𝑣 ∈ 𝑆 is perfect, and is internally almost perfect if every vertex 𝑣 ∈ 𝑆 is almost perfect; for brevity we say that an internally almost perfect set is an ap-set. Let 𝜃 ap (𝐺) and Θap (𝐺) equal the minimum and maximum cardinality, respectively, of a maximal ap-set in 𝐺.

Section 2.5. Domination in Terms of Perfection in Graphs

43

Definition 2.58 A set 𝑆 ⊆ 𝑉 is externally perfect if every vertex in 𝑉 \ 𝑆 is perfect, and is externally almost perfect if every vertex in 𝑉 \ 𝑆 is either perfect or adjacent to a perfect vertex; for brevity we say that an externally almost perfect set is an eap-set. Let 𝜃 ap (𝐺) and Θap (𝐺) equal the minimum and maximum cardinality, respectively, of a minimal eap-set in 𝐺. Similarly, a set 𝑆 is completely perfect if every vertex 𝑣 ∈ 𝑉 is perfect, that is, if 𝑆 is both internally and externally perfect. In the graph in Figure 2.6, given in [439], a vertex labeled “p” is perfect, while a vertex labeled “ap” is almost perfect with respect to the three red vertices forming a set 𝑆, two of which are almost perfect while the third is perfect. This set 𝑆 is also externally almost perfect since every vertex in 𝑉 \ 𝑆 is either perfect with respect to 𝑆 or adjacent to a perfect vertex. p p

ap p

ap ap

ap

p ap

p p

ap ap

Figure 2.6 A perfect neighborhood set

Definition 2.59 A set 𝑆 ⊆ 𝑉 is a perfect neighborhood set if every vertex 𝑣 ∈ 𝑉 is either perfect or is adjacent to a perfect vertex (see [191, 284, 310, 440]). The perfect neighborhood number 𝜃 (𝐺) is the minimum cardinality of a perfect neighborhood set in 𝐺, while the upper perfect neighborhood number Θ(𝐺) is the maximum cardinality ap ap of a perfect neighborhood set in 𝐺. Let 𝜃 p (𝐺) and Θp (𝐺) equal the minimum and maximum cardinality, respectively, of an independent perfect neighborhood set in 𝐺. The following somewhat surprising theorem was first proved in [310]. We give a proof of this result in Chapter 14, where a more detailed discussion of the upper domination number is given. Theorem 2.60 ([310]) For any graph 𝐺, Γ(𝐺) = Θ(𝐺). Definition 2.61 A set 𝑆 ⊆ 𝑉 is an eap irredundant set, eap dominating set, or an eap independent set if it is a maximal irredundant, minimal dominating, or maximal independent set that is also an eap-set. Thus, every vertex 𝑣 ∈ 𝑉 \ 𝑆 is either perfect or is adjacent to a perfect vertex. Let irap (𝐺), 𝛾 ap (𝐺), 𝑖 ap (𝐺), 𝛼ap (𝐺) Γap (𝐺), and IRap (𝐺) denote the minimum and maximum cardinalities of such sets. Given these definitions, we can relate them to independent, dominating, and irredundant sets. For example, the concept of a set 𝑆 being internally perfect is equivalent to the concept of a set being independent.

44

Chapter 2. Fundamentals of Domination

Proposition 2.62 ([439]) A set 𝑆 is internally perfect if and only if it is independent. Proof A vertex 𝑢 ∈ 𝑆 is perfect if and only if |N[𝑢] ∩ 𝑆| = 1, that is, it is an isolated vertex in the induced subgraph 𝐺 [𝑆]. Thus, if every vertex in 𝑆 is perfect, then 𝑆 is an independent set. Conversely, if 𝑆 is an independent set, then clearly every vertex 𝑣 ∈ 𝑆 satisfies the condition that |N[𝑣] ∩ 𝑆| = 1. Therefore, 𝑆 is internally perfect. ap

Corollary 2.63 ([439]) For any graph 𝐺, 𝜃 p (𝐺) ≤ 𝑖(𝐺) = 𝑖 ap (𝐺). Proof We first show that 𝑖(𝐺) = 𝑖 ap (𝐺). Let 𝑆 be a maximal independent set of minimum cardinality, and so |𝑆| = 𝑖(𝐺). By Proposition 2.62, the set 𝑆 is internally perfect. By definition, 𝑆 is also a minimal dominating set. Thus, every vertex in 𝑆 is perfect, and every vertex in 𝑉 \ 𝑆 is adjacent to a perfect vertex. Thus, 𝑆 is an internally perfect set whose complement 𝑉 \ 𝑆 is an ap-set, and therefore, 𝑖 ap (𝐺) ≤ |𝑆| = 𝑖(𝐺). Conversely, since every 𝑖 ap -set of 𝐺 is maximal independent, it is therefore an independent dominating set, implying that 𝑖(𝐺) ≤ 𝑖 ap (𝐺). ap The fact that 𝜃 p (𝐺) ≤ 𝑖(𝐺) follows from the observation that every maximal independent set is an independent perfect neighborhood set. ap

Corollary 2.64 ([439]) For any graph 𝐺, 𝛼(𝐺) = 𝛼ap (𝐺) = Θp (𝐺). Proof As discussed previously in this chapter, every 𝛼-set 𝑆 of 𝐺 is both a maximal independent set and a minimal dominating set. Therefore, from Proposition 2.62, 𝑆 is an internally perfect set and an eap-set, since every vertex in 𝑉 \ 𝑆 is adjacent to a perfect vertex in 𝑆. It is therefore an independent perfect neighborhood set, and ap ap thus, 𝛼(𝐺) ≤ Θp (𝐺). But every Θp -set of 𝐺 is an independent set, and therefore by ap definition, Θp (𝐺) ≤ 𝛼(𝐺). Similarly, every 𝛼-set 𝑆 of 𝐺 is a maximal independent set that is also externally almost perfect, since every vertex in 𝑉 \ 𝑆 is adjacent to a perfect vertex in 𝑆. Thus, 𝛼(𝐺) ≤ 𝛼ap (𝐺). But every 𝛼ap -set of 𝐺 is an independent set, and therefore by definition, 𝛼ap (𝐺) ≤ 𝛼(𝐺). The next results follow directly from the definitions. Proposition 2.65 ([439]) A set 𝑆 is externally perfect if and only if 𝑆 is a perfect dominating set. Proposition 2.66 ([439]) A set 𝑆 is completely perfect if and only if 𝑆 is an efficient dominating set. We remark that an efficient dominating set is also called a perfect code in the literature, and efficient dominating sets will be covered in Chapter 9. Next, we show that the concept of being internally almost perfect (ap) is equivalent to the concept of being irredundant. Proposition 2.67 ([439]) A set 𝑆 is internally almost perfect if and only if 𝑆 is irredundant. Proof If a set 𝑆 is internally almost perfect, then every vertex 𝑢 ∈ 𝑆 is either perfect or adjacent to a perfect vertex. Either 𝑢 is an isolated vertex in 𝐺 [𝑆], in which case it

Section 2.5. Domination in Terms of Perfection in Graphs

45

is perfect and is its own 𝑆-private neighbor or 𝑢 is adjacent to a perfect vertex 𝑤. Thus, 𝑤 ∈ 𝑉 \ 𝑆 and |N[𝑤] ∩ 𝑆| = |{𝑢}| = 1, and so 𝑤 is an 𝑆-external private neighbor of 𝑢. Therefore, every vertex 𝑢 ∈ 𝑆 has an 𝑆-private neighbor and hence 𝑆 is irredundant. Conversely, if 𝑆 is irredundant, then every vertex 𝑢 ∈ 𝑆 is either its own 𝑆-private neighbor, in which case 𝑢 is perfect or has an 𝑆-external private neighbor 𝑤. But in this case 𝑤 is perfect and therefore 𝑢 is adjacent to a perfect vertex. Therefore, 𝑆 is internally almost perfect. Corollary 2.68 ([439]) For any graph 𝐺, ir(𝐺) = 𝜃 ap (𝐺) ≤ Θap (𝐺) = IR(𝐺). The Domination Chain, restated below as Theorem 2.69, can now be considerably expanded in terms of the concept of perfection. Theorem 2.69 ([196]) For any graph 𝐺, ir(𝐺) ≤ 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺) ≤ IR(𝐺). Theorem 2.70 For any graph 𝐺, the following system of inequalities holds: ap

ap

𝜃 ap (𝐺) ≤ 𝜃 (𝐺) ≤ 𝜃 p (𝐺) ≤ Θp (𝐺) ≤ Θ(𝐺) ≤ Θap (𝐺) =

=

=

≤

≤

𝜃 ap (𝐺) = ir(𝐺) ≤ 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺) ≤ IR(𝐺) ≥

≥

=

=

≤

≤

𝜃 ap (𝐺) ≤ irap (𝐺) ≤ 𝛾 ap (𝐺) ≤ 𝑖 ap (𝐺) ≤ 𝛼ap (𝐺) ≤ Γ(𝐺) ≤ IRap (𝐺) ≥ 𝜃 (𝐺) ap

Let 𝛾d (𝐺) equal the minimum cardinality of a dominating set that is externally almost perfect. Let 𝛾 ap (𝐺) equal the minimum cardinality of a minimal dominating set that is externally almost perfect. These two parameters lead to the following refinement. ap

Proposition 2.71 ([439]) For any graph 𝐺, 𝛾(𝐺) ≤ 𝛾d (𝐺) ≤ 𝛾 ap (𝐺) ≤ 𝑖(𝐺). As given in [439], the fact that each of these inequalities can be strict is illustrated by the unicyclic graph 𝐺 in Figure 2.7. For this graph, 𝛾(𝐺) = 4 and {𝑣 1 , 𝑣 3 , 𝑣 4 , 𝑣 6 } ap ap is a 𝛾-set of 𝐺, 𝛾d (𝐺) = 6 and {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 , 𝑣 5 , 𝑣 6 } is a 𝛾d -set of 𝐺, 𝛾 ap (𝐺) = 8 and {𝑣 1 , 𝑣 6 , 𝑣 7 , 𝑣 8 , 𝑣 9 , 𝑣 10 , 𝑣 11 , 𝑣 12 } is a 𝛾 ap -set of 𝐺, and 𝑖(𝐺) = 9 and {𝑣 1 , 𝑣 4 , 𝑣 7 , 𝑣 8 , 𝑣 9 , 𝑣 13 , 𝑣 14 , 𝑣 15 , 𝑣 16 } is an 𝑖-set of 𝐺. The concept of perfection in graphs provides a framework for unifying the concepts of independent sets, dominating sets, irredundant sets, perfect and efficient dominating sets, and perfect neighborhood sets. For example: (a) A set is independent if and only if it is an internally perfect set. (b) The independence parameters 𝑖(𝐺) and 𝛼(𝐺) can be expressed as maximal independent sets whose complements are internally almost perfect, that is, 𝑖(𝐺) = 𝑖 ap (𝐺) and 𝛼(𝐺) = 𝛼ap (𝐺). (c) A set is an irredundant set if and only if it is an internally almost perfect set.

Chapter 2. Fundamentals of Domination

46 𝑣 20 𝑣 19

𝑣7 𝑣2

𝑣8

𝑣 18 𝑣1

𝑣3

𝑣9

𝑣6

𝑣4

𝑣 10

𝑣 17 𝑣 16 𝑣 15 𝑣 14 𝑣 13

𝑣5

𝑣 11 𝑣 12

ap

Figure 2.7 𝛾(𝐺) < 𝛾d 𝐺 < 𝛾 ap (𝐺) < 𝑖(𝐺)

(d) The parameters ir(𝐺) and IR(𝐺) can be expressed in terms of internally almost perfect sets, namely, ir(𝐺) = 𝜃 ap (𝐺) and IR(𝐺) = Θap (𝐺). (e) Theorem 2.60, that Θ(𝐺) = Γ(𝐺), establishes an equality between two seemingly unrelated parameters. This result is now clearer. In particular, the inequality chain: 𝛼(𝐺) ≤ Γ(𝐺) ≤ IR(𝐺) can now be stated equivalently ap ap as: Θp (𝐺) ≤ Θ(𝐺) ≤ Θap (𝐺), since Θp (𝐺) = 𝛼(𝐺), Θ(𝐺) = Γ(𝐺), and Θap (𝐺) = IR(𝐺). (f) An expanded inequality chain exists between the domination and independence parameters: ap

𝛾(𝐺) ≤ 𝛾d (𝐺) ≤ 𝛾 ap (𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γap (𝐺) ≤ Γ(𝐺).

2.6

Ore’s Lemmas and Their Implications

The property of being a dominating set is superhereditary, that is, every superset of a dominating set in a graph 𝐺 is also a dominating set of 𝐺. Thus, by Proposition 2.2, a dominating set 𝐷 is minimal if and only if for every vertex 𝑣 ∈ 𝐷, the set 𝐷 \ {𝑣} is not a dominating set. Therefore, it suffices, in order to show minimality of a dominating set 𝐷, for us only to consider 𝐷 \ {𝑣} for each vertex 𝑣 ∈ 𝐷 rather than all possible nonempty subsets of 𝐷. In 1962 Ore [622] proved the following properties of minimal dominating sets in graphs. Lemma 2.72 ([622]) A dominating set 𝐷 in a graph 𝐺 is a minimal dominating set of 𝐺 if and only if every vertex 𝑣 ∈ 𝐷 either is an internal private neighbor or has an external private neighbor, that is, if and only if ipn[𝑣, 𝐷] ≠ ∅ or epn[𝑣, 𝐷] ≠ ∅. Proof Suppose, firstly, that 𝐷 is a minimal dominating set of a graph 𝐺. Let 𝑣 be an arbitrary vertex of 𝐷. By the minimality of 𝐷, the set 𝐷 \ {𝑣} is not a dominating set of 𝐺. Let 𝑤 be a vertex of 𝐺 not dominated by 𝐷 \ {𝑣}. If 𝑤 ∈ 𝐷, then 𝑤 = 𝑣, implying that 𝑣 is isolated in 𝐺 [𝐷] and therefore ipn[𝑣, 𝐷] = {𝑣}. If 𝑤 ∈ 𝑉 \ 𝐷, then 𝑤 is adjacent to 𝑣 but to no other vertex of 𝐷, implying that 𝑤 ∈ epn[𝑣, 𝐷]. Thus, ipn[𝑣, 𝐷] ≠ ∅ or epn[𝑣, 𝐷] ≠ ∅. Conversely, if ipn[𝑣, 𝐷] ≠ ∅ or epn[𝑣, 𝐷] ≠ ∅ for

Section 2.6. Ore’s Lemmas and Their Implications

47

every vertex 𝑣 ∈ 𝐷, then the set 𝐷 \ {𝑣} is not a dominating set of 𝐺 for each such vertex 𝑣. Lemma 2.73 ([622]) If 𝐺 = (𝑉, 𝐸) is an isolate-free graph, then the complement 𝑉 \ 𝐷 of any minimal dominating set 𝐷 is a dominating set. Proof Let 𝐷 be a minimal dominating set of 𝐺. We show that every vertex in 𝐷 has a neighbor in 𝑉 \ 𝐷. Let 𝑣 be an arbitrary vertex of 𝐷. Since 𝐺 is isolate-free, the vertex 𝑣 has at least one neighbor. By Lemma 2.72, ipn[𝑣, 𝐷] ≠ ∅ or epn[𝑣, 𝐷] ≠ ∅. If ipn[𝑣, 𝐷] ≠ ∅, then 𝑣 is isolated in 𝐺 [𝐷] and therefore all neighbors of 𝑣 belong to 𝑉 \ 𝐷. If epn[𝑣, 𝐷] ≠ ∅, then 𝑣 has a neighbor in 𝑉 \ 𝐷. Thus, 𝑉 \ 𝐷 dominates every vertex in 𝐷 and is a dominating set of 𝐺. As a consequence of Lemma 2.73, we have the following result. Corollary 2.74 ([622]) In any nontrivial connected graph 𝐺 = (𝑉, 𝐸), there is a partition of the vertex set 𝑉 into two sets, each of which is a dominating set; furthermore, one of the sets can be chosen to be either (i) a minimal dominating set, (ii) a 𝛾-set, (iii) a Γ-set, (iv) a maximal independent set, (v) an 𝑖-set, or (vi) an 𝛼-set. It was Ore’s Lemma 2.73 that prompted Cockayne and Hedetniemi [194] to introduce the concept of the domatic number of a graph in 1977. Definition 2.75 The domatic number dom(𝐺) of a graph 𝐺 equals the maximum order 𝑘 of a vertex partition 𝜋 = {𝑉1 , 𝑉2 , . . . , 𝑉𝑘 } such that each subset 𝑉𝑖 is a dominating set of 𝐺. As an immediate consequence of Ore’s Lemma 2.73, the domatic number of a nontrivial connected graph is at least 2. Corollary 2.76 ([622]) If 𝐺 is a connected graph of order 𝑛 ≥ 2, then dom(𝐺) ≥ 2. We shall see more implications of Ore’s Lemmas in Chapter 4 when we discuss general bounds on the domination number, in Chapter 6 when we discuss upper bounds on the domination number, and in Chapter 12 when we discuss dominating partitions.

Chapter 3

Complexity and Algorithms for Domination in Graphs 3.1 Introduction In this chapter, we provide an overview of some of the core results on NP-completeness and algorithms for domination, independent domination, and total domination in graphs. First, we consider the following general question: given an arbitrary graph 𝐺, how difficult is it to determine, or compute, the value of any of the following six basic domination parameters, which satisfy the following well-known inequalities: 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺) and 𝛾(𝐺) ≤ 𝛾t (𝐺) ≤ Γt (𝐺). In the best cases, there is either a simple method to determine the value of any one of these domination parameters, or at least a polynomial algorithm will exist which enables these values to be determined with relatively little computer time. Unfortunately, these best cases are quite rare. It is well known that all of these problems are NP-complete when the input is an arbitrary graph. This means that the general expectation is that it will require an amount of time that is exponential, as a function of the order and size of the graph, in order to compute any of these domination numbers for an arbitrary graph. Therefore, the general expectation is that we can compute these numbers in a reasonable amount of time only for graphs of relatively small order, say 𝑛 ≤ 100, with a few notable exceptions, of course. If, on the other hand, we restrict the types of graphs for which we would like to compute the value of these parameters, then we can expect some degree of success. Indeed, a significant amount of research has been dedicated to finding classes of graphs for which these problems can be solved in polynomial time. An almost equal amount of time has been spent determining classes of graphs for which these problems are NP-complete. As a very simple example, it is known that a simple linear algorithm © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_3

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Chapter 3. Complexity and Algorithms for Domination in Graphs

exists for computing the value of 𝛾(𝑇) when the input is an arbitrary tree 𝑇. But the class of trees is clearly a proper subclass of the class of bipartite graphs, and it is known that the domination problem is NP-complete for arbitrary bipartite graphs. The recognition of this fact has therefore led researchers to find classes of graphs that properly contain trees as a subclass, but which are themselves properly contained within the class of bipartite graphs, such that the domination problem can either be computed in polynomial time for this class or that the problem is still NP-complete for this subclass of bipartite graphs. The possibilities for research, either in finding polynomial algorithms or NPcompleteness results in this area, are almost unlimited. In the book, Graph Classes, A Survey by Brandstädt et al. [102], definitions are given of close to 150 different classes of graphs that have been studied. And in the paper, An Annotated Glossary of Graph Theory Parameters with Conjectures by Gera et al. [333], definitions are given of some 300 graph theory parameters, including approximately 80 parameters related to domination in graphs. Thus, from the perspective of possible algorithms and NP-completeness results related to domination in graphs, there are at least 150 × 80 = 12,000 possible combinations. The possibilities are even larger than this because one can construct: (i) an exact polynomial-time sequential algorithm, (ii) an exact exponential-time sequential algorithm, (iii) an exact, parallel 𝑛-processor algorithm, (iv) a sequential approximation algorithm, (v) a self-stabilizing algorithm, (vi) a parallel distributed algorithm, (vii) a greedy algorithm using heuristics, (viii) a mixed integer-linear program, (ix) a fixed parameter tractable algorithm, or (x) a genetic algorithm. And for each type of algorithm, the vertices and/or edges can be weighted or unweighted, the edges can be directed or undirected, and there may or may not be loops and multiple edges. In this chapter we present, without attempting to be comprehensive, a brief review of some of the classes of graphs for which domination algorithms or NPcompleteness proofs have been constructed, focusing only the basic six domination parameters 𝛾(𝐺), 𝑖(𝐺), 𝛼(𝐺), Γ(𝐺), 𝛾t (𝐺), and Γt (𝐺).

3.2

Brief Review of NP-Completeness

Basically, NP-completeness has to do with three classes of computational problems: the class P, the class NP, and the class NPc. A computation problem is in the class P if there exists an algorithm for solving any input to problem in O (𝑛 𝑘 ) time, for some fixed integer 𝑘 ≥ 1, where 𝑛 denotes the length of the input. Several common examples of computational problems that are known to be in the class P are: (a) finding a maximum or minimum of a set of 𝑛 integers (b) computing the sum or product of 𝑛 integers (c) sorting a set of 𝑛 integers (d) finding a shortest path between two vertices in a connected graph (e) finding a maximum matching in a graph (f) finding the maximum or minimum degree of a vertex in a graph (g) finding a spanning tree of a graph

Section 3.2. Brief Review of NP-Completeness

51

deciding if a graph is connected deciding if a graph is bipartite computing the diameter of a graph deciding if a graph is planar deciding if two trees, planar graphs, interval graphs, or permutation graphs are isomorphic. A computation problem is in the class NP if it can be solved in polynomial time by a nondeterministic Turing machine. Thus, the letters NP stand for Nondeterministic Polynomial time. So, what is a Turing machine? What is a nondeterministic Turing machine? Without getting overly technical, let us illustrate this with an intuitive example. Suppose we are given an arbitrary graph 𝐺 and we wish to know if 𝐺 has a dominating set 𝑆 of cardinality at most 𝑘 for some 𝑘 ≥ 1. Thus, we are given as input to this nondeterministic algorithm: (i) a graph 𝐺, and (ii) the question: does 𝐺 have a dominating set 𝑆 of cardinality at most 𝑘? A nondeterministic algorithm, or nondeterministic Turing machine, has the ability, and is granted permission, to examine in polynomial time, one-by-one, every vertex in the graph 𝐺 and make a ‘guess’ as to whether this vertex is in a dominating set of cardinality at most 𝑘. Having reviewed all vertices, it selects a set 𝑆 of cardinality at most 𝑘. It then must be able to determine, again in polynomial time, whether this set is a dominating set. If it is a dominating set, then the nondeterministic algorithm answers the given question, ‘yes.’ We say that a graph 𝐺 is a ‘yes’ instance to this problem if it does have a dominating set of cardinality at most 𝑘, otherwise, it is a ‘no’ instance. The nondeterministic algorithm is required to say ‘yes’ to at least one execution if the graph is a ‘yes’ instance, and never to say ‘yes’ to a ‘no’ instance, and never take more than a polynomial amount of time for any instance. An equivalent, and perhaps easier, formulation of nondeterministic algorithms can be given in terms of polynomial-time verification. All that is required is to have a deterministic algorithm, called a verification algorithm, which can be given an instance of the problem, a graph 𝐺, and a set 𝑆, called a candidate solution, and be able to verify or decide in polynomial time whether 𝑆 is a solution for the graph 𝐺. In this view, the class NP is the class of all problems which can be verified in polynomial time. This leaves us to define the class NPc of NP-complete problems. We say that a computation problem 𝐴 is polynomial-time reducible to a computation problem 𝐵, we can write this as 𝐴 → 𝐵, if (i) there exists a function 𝑓 which maps any instance 𝐼 of problem 𝐴 to an instance 𝑓 (𝐼) of problem 𝐵, such that 𝐼 is a ‘yes’ instance of problem 𝐴 if and only if 𝑓 (𝐼) is a ‘yes’ instance of problem 𝐵, and (ii) for any instance 𝐼 of problem 𝐴, the corresponding instance 𝑓 (𝐼) of problem 𝐵 can be constructed in polynomial time. Note that it follows from this definition, that if problem 𝐴 is polynomial-time reducible to problem 𝐵, and problem 𝐵 can be solved in polynomial time, then problem 𝐴 can also be solved in polynomial time. In order to solve an instance 𝐼 of problem 𝐴, simply construct in polynomial time the instance 𝑓 (𝐼) of problem 𝐵 and (h) (i) (j) (k) (l)

Chapter 3. Complexity and Algorithms for Domination in Graphs

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use the polynomial algorithm for problem 𝐵 to determine whether 𝑓 (𝐼) is a ‘yes’ or ‘no’ instance. We define a problem 𝐴 to be NP-complete if (i) 𝐴 is in the class NP, which can be shown by constructing a polynomial verification algorithm for 𝐴, and (ii) for any problem 𝑋 ∈ NPc, 𝑋 → 𝐴. We say that a problem is NP-hard if it satisfies condition (ii) but not necessarily condition (i). It is easy to see that the relation 𝐴 → 𝐵 is transitive. Thus, if 𝐴 → 𝐵 and 𝐵 → 𝐶, then 𝐴 → 𝐶. This means that an algorithm for solving problem 𝐶 can be used not only to solve problem 𝐵 but also to solve problem 𝐴. And it follows from this that any algorithm for solving a problem in NPc can be used to solve all problems in NPc. Thus, in order to show that a computation problem 𝐴 is NP-complete, that is, that 𝐴 ∈ NPc, one must: (i) Show that 𝐴 ∈ NP, that is, construct a polynomial-time verification algorithm for 𝐴. (ii) Find a known NP-complete problem 𝑋 ∈ NPc and show that 𝑋 → 𝐴, that is, an algorithm for solving 𝐴 can be used to solve the known NP-complete problem 𝑋. We illustrate this process of showing that a problem is NP-complete in the next section.

3.3

NP-Completeness of Domination, Independent Domination, and Total Domination

In this section, we consider the NP-completeness of domination, independent domination and total domination. NP-completeness results are presented in Section 3.3.1 for arbitrary graphs and Section 3.3.2 for bipartite graphs. A summary of NP-completeness results for some families of graphs is presented in Section 3.3.3.

3.3.1

NP-Completeness Results for Arbitrary Graphs

To the best of our knowledge, the first proof of the NP-completeness of the domination problem was due to David Johnson at a graph theory conference held in Qualicum Beach, Vancouver Island, British Columbia sometime during 1975–1976 [510], using a simple reduction, shown below, from the well-known NP-complete problem called 3SAT. This result was subsequently stated in the NP-completeness book by Garey and Johnson [325] as [GT2] DOMINATING SET. The basic statement of this problem, as a decision problem, is the following: DOMINATING SET

Instance: A graph 𝐺 = (𝑉, 𝐸) and a positive integer 𝑘. Question: Does 𝐺 have a dominating set of cardinality at most 𝑘? Theorem 3.1 ([510]) DOMINATING SET is NP-complete.

Section 3.3. NP-Completeness of Domination Parameters

53

Proof We must first show that DOMINATING SET ∈ NP, that is, construct a polynomial-time verification algorithm for DOMINATING SET. This part of the proof is straightforward. Given a graph 𝐺 and a set 𝑆 of vertices, one can easily check in polynomial time to see that |𝑆| ≤ 𝑘 and that 𝑆 is or is not a dominating set of 𝐺, that is, check to see that for all 𝑣 ∈ 𝑉 \ 𝑆, N(𝑣) ∩ 𝑆 ≠ ∅. Now we must find a known NP-complete problem 𝑋 ∈ NPc and show that 𝑋 → DOMINATING SET. Johnson chose one of the most commonly known NPcomplete problems, 3-SATISFIABILITY (or 3SAT). 3SAT

Instance: A set 𝑉 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 } of 𝑛 Boolean variables and a set 𝐶 = {𝐶1 , 𝐶2 , . . . , 𝐶𝑚 } of 𝑚 clauses, each being a disjunction of three variables or their complements, such as 𝐶1 = (𝑣 1 ∨ 𝑣 2 ∨ 𝑣 5 ). Question: Does 𝐶 have a satisfying truth assignment, that is, a function 𝑓 : 𝑉 → {TRUE, FALSE} such that at least one variable or its complement in each clause is assigned the value TRUE? Thus, the clause 𝐶1 = (𝑣 1 ∨ 𝑣 2 ∨ 𝑣 5 ) will be true if either 𝑓 (𝑣 1 ) = TRUE or 𝑓 (𝑣 2 ) = FALSE or 𝑓 (𝑣 5 ) = TRUE. We must construct a polynomial time reduction from an instance of 3SAT to an instance of DOMINATING SET. We do this as follows. Given an instance 𝑉 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 } and 𝐶 = {𝐶1 , 𝐶2 , . . . , 𝐶𝑚 } of 3SAT, we construct an instance of DOMINATING SET, that is, a graph 𝐺 (𝑉, 𝐶) and a positive integer 𝑘 = 𝑛 (see Figure 3.1). 𝑥1 𝑣1

𝑐1

𝑥3

𝑥2 𝑣1

𝑣2

𝑐2

𝑣2

𝑣3

𝑐3

𝑥5

𝑥4 𝑣3

𝑣4

𝑣4

𝑣5

𝑣5

𝑐4

Figure 3.1 Reduction from 3SAT to DOMINATING SET For each variable 𝑣 𝑖 ∈ 𝑉, construct a triangle with the top vertex labeled 𝑥𝑖 and the bottom two vertices labeled 𝑣 𝑖 and 𝑣 𝑖 . For each clause 𝐶 𝑗 ∈ 𝐶, construct a single vertex labeled 𝑐 𝑗 . Finally, for each clause, such as 𝐶1 = (𝑣 1 ∨ 𝑣 2 ∨ 𝑣 3 ), add the three edges 𝑐 1 𝑣 1 , 𝑐 1 𝑣 2 , and 𝑐 1 𝑣 3 , as shown in Figure 3.1. We must show that (i) 𝑉, 𝐶 is a ‘yes’ instance of 3SAT if and only if the graph 𝐺 (𝑉, 𝐶) so constructed with the positive integer 𝑘 = 𝑛 is a ‘yes’ instance of

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Chapter 3. Complexity and Algorithms for Domination in Graphs

DOMINATING SET, and (ii) that for any instance 𝑉, 𝐶 of 3SAT, we can construct the instance 𝐺 (𝑉, 𝐶) and 𝑘 in polynomial time. But this is easy to see. The size of an instance of 3SAT is O (𝑛 + 𝑚), that is, 𝑛 Boolean variables and 𝑚 clauses, each clause containing three variables. The size of the constructed instance of DOMINATING SET is 3𝑛 + 𝑚 vertices and 3𝑛 + 3𝑚 edges. Thus, the size of an instance of DOMINATING SET is at most a constant times the size of an instance of 3SAT. This brings us to (i). Assume that an instance 𝑉, 𝐶 of 3SAT has a satisfying truth assignment 𝑓 : 𝑉 → {TRUE, FALSE}. We construct a set 𝑆 of the graph 𝐺 (𝑉, 𝐶) as follows: if 𝑓 (𝑣 𝑖 ) = TRUE, then place vertex 𝑣 𝑖 ∈ 𝑆, and if 𝑓 (𝑣 𝑗 ) = FALSE, then place vertex 𝑣 𝑗 ∈ 𝑆. Notice that the set 𝑆 so constructed contains exactly one vertex from each of the 𝑛 triangles. Thus, each vertex in a triangle is either in 𝑆 or is adjacent to exactly one vertex in 𝑆. In addition, each clause vertex 𝑐 𝑖 is adjacent to at least one vertex in 𝑆 because 𝑓 is a satisfying truth assignment, which means that at least one variable in each clause is assigned the value TRUE. In other words, if 𝑓 (𝑣 𝑖 ) = TRUE, then vertex 𝑣 𝑖 ∈ 𝑆, and if 𝑓 (𝑣 𝑖 ) = FALSE, then 𝑣 𝑖 ∈ 𝑆. Thus, 𝑆 is a dominating set of the graph 𝐺 (𝑉, 𝐶) and the cardinality of 𝑆 is exactly 𝑛. Hence, if 𝑉, 𝐶 is a ‘yes’ instance of 3SAT, then the graph 𝐺 (𝑉, 𝐶) is a ‘yes’ instance of DOMINATING SET. Conversely, assume that 𝐺 (𝑉, 𝐶) is a ‘yes’ instance of DOMINATING SET, that is, 𝐺 has a dominating set 𝑆 such that |𝑆| ≤ 𝑘 = 𝑛. Since 𝑆 is a dominating set, it must contain at least one vertex in each of the 𝑛 triangles. Hence, |𝑆| ≥ 𝑛. If a vertex 𝑥 𝑗 ∈ 𝑆, then we can replace it with either 𝑣 𝑗 or 𝑣 𝑗 and still have a dominating set. Thus, we may assume 𝑥 𝑗 ∉ 𝑆 for 𝑗 ∈ [𝑛]. Therefore, |𝑆| = 𝑛 and the vertices in 𝑆 must dominate all clause vertices 𝑐 𝑖 . Hence, if a clause vertex 𝑐 𝑖 is dominated by a triangle vertex 𝑣 𝑗 , define a truth assignment such that 𝑓 (𝑣 𝑗 ) = TRUE. Similarly, if a clause vertex is dominated by a triangle vertex 𝑣 𝑗 , define 𝑓 (𝑣 𝑗 ) = FALSE. This produces a satisfying truth assignment for the 3SAT instance 𝑉, 𝐶.

Notice that the dominating set constructed in the NP-completeness proof of DOMINATING SET is an independent dominating set. This provides us as a corollary

a second NP-completeness proof to the following decision problem. INDEPENDENT DOMINATING SET

Instance: A graph 𝐺 = (𝑉, 𝐸) and a positive integer 𝑘 Question: Does 𝐺 have a independent dominating set of cardinality at most 𝑘? Corollary 3.2 ([510]) INDEPENDENT DOMINATING SET is NP-complete. Proof The proof is identical, word-for-word, to the proof of the NP-completeness of DOMINATING SET, with the single exception that every occurrence of DOMINATING SET or dominating set is replaced by INDEPENDENT DOMINATING SET. A very simple change in the construction of the graph 𝐺 (𝑉, 𝐶) in the NP-completeness proof for DOMINATING SET will enable us construct an NP-completeness proof for the following decision problem. Recall that we abbreviate a total dominating set by TD-set.

Section 3.3. NP-Completeness of Domination Parameters

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TOTAL DOMINATING SET

Instance: A graph 𝐺 = (𝑉, 𝐸) and a positive integer 𝑘. Question: Does 𝐺 have a TD-set of cardinality at most 𝑘? Corollary 3.3 ([510]) TOTAL DOMINATING SET is NP-complete. Proof The proof is the same as the proof of the NP-completeness of DOMINATING SET, with the following changes. We must construct a polynomial time reduction from an instance of 3SAT to an instance of TOTAL DOMINATING SET. We do this as follows. Given an instance 𝑉 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 } and 𝐶 = {𝐶1 , 𝐶2 , . . . , 𝐶𝑚 } of 3SAT, we construct an instance of TOTAL DOMINATING SET, that is, a graph 𝐺 (𝑉, 𝐶) and a positive integer 𝑘 = 2𝑛. For each variable 𝑣 𝑖 ∈ 𝑉, construct a triangle with the top vertex labeled 𝑥𝑖 and the bottom two vertices labeled 𝑣 𝑖 , 𝑣 𝑖 . To this triangle add a fourth vertex 𝑦 𝑖 , a leaf which is adjacent only to vertex 𝑥𝑖 . For each clause 𝐶 𝑗 ∈ 𝐶, construct a single vertex labeled 𝑐 𝑗 . Finally, for each clause, such as 𝐶1 = (𝑣 1 ∧ 𝑣 2 ∧ 𝑣 3 ), add the three edges 𝑐 1 𝑣 1 , 𝑐 1 𝑣 2 , and 𝑐 1 𝑣 3 . We must show (i) that 𝑉, 𝐶 is a ‘yes’ instance of 3SAT if and only if the graph 𝐺 (𝑉, 𝐶) so constructed with the positive integer 𝑘 = 2𝑛 is a ‘yes’ instance of TOTAL DOMINATING SET. We must also show (ii) that for any instance 𝑉, 𝐶 of 3SAT, we can construct the instance 𝐺 (𝑉, 𝐶) and 𝑘 = 2𝑛 in polynomial time. But this is easy to see. The size of an instance of 3SAT is O (𝑛 + 𝑚), that is, 𝑛 Boolean variables and 𝑚 clauses, each containing three variables. The size of the constructed instance 𝐺 (𝑉, 𝐶) of TOTAL DOMINATING SET is 4𝑛 + 𝑚 vertices and 4𝑛 + 3𝑚 edges. Thus, the size of an instance of TOTAL DOMINATING SET is at most a constant times the size of an instance of 3SAT. This brings us to (i). Assume that an instance 𝑉, 𝐶 has a satisfying truth assignment 𝑓 : 𝑉 → {TRUE, FALSE}. We construct a set 𝑆 of the graph 𝐺 (𝑉, 𝐶) as follows: if 𝑓 (𝑣 𝑖 ) = TRUE, then place vertex 𝑣 𝑖 ∈ 𝑆, and if 𝑓 (𝑣 𝑗 ) = FALSE, then place vertex 𝑣 𝑗 ∈ 𝑆. In addition, place into the set 𝑆 all 𝑛 vertices of the form 𝑥𝑖 . Thus, |𝑆| = 2𝑛. Notice that the set 𝑆 so constructed contains exactly two vertices from each of the 𝑛 triangles. Thus, each vertex in a triangle is either in 𝑆 and has exactly one neighbor in 𝑆, or is adjacent to exactly two vertices in 𝑆. Each leaf 𝑦 𝑖 is adjacent to 𝑥𝑖 ∈ 𝑆. In addition, each clause vertex 𝑐 𝑖 is adjacent to at least one vertex in 𝑆 because 𝑓 is a satisfying truth assignment, which means that at least one variable in each clause is assigned the value TRUE, which means that if 𝑓 (𝑣 𝑖 ) = TRUE, then vertex 𝑣 𝑖 ∈ 𝑆, and if 𝑓 (𝑣 𝑖 ) = FALSE, then 𝑣 𝑖 ∈ 𝑆. Thus, 𝑆 is a TD-set of the graph 𝐺 (𝑉, 𝐶) and the cardinality of 𝑆 is exactly 2𝑛. Thus, if 𝑉, 𝐶 is a ‘yes’ instance of 3SAT, then the graph 𝐺 (𝑉, 𝐶) is a ‘yes’ instance of TOTAL DOMINATING SET. Conversely, assume that 𝐺 (𝑉, 𝐶) is a ‘yes’ instance of TOTAL DOMINATING SET, that it has a TD-set 𝑆 of cardinality |𝑆| ≤ 2𝑛. Since 𝑆 is a TD-set, it must contain at least two vertices in each of the 𝑛 subgraphs consisting of a triangle with an attached leaf. Furthermore, if 𝑆 were to contain the pair of vertices 𝑥𝑖 , 𝑦 𝑖 , it could be

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replaced by either the pair 𝑥𝑖 , 𝑣 𝑖 or the pair 𝑥𝑖 , 𝑣 𝑖 . Thus, we can assume that 𝑆 does not contain any 𝑦 𝑖 vertex. Therefore, |𝑆| ≥ 2𝑛. Consequently, |𝑆| = 2𝑛 and the vertices in 𝑆 must dominate all clause vertices 𝑐 𝑖 . Hence, if a clause vertex 𝑐 𝑖 is dominated by a triangle vertex 𝑣 𝑗 , define a truth assignment such that 𝑓 (𝑣 𝑗 ) = TRUE. Similarly, if a clause vertex is dominated by a triangle vertex 𝑣 𝑗 define 𝑓 (𝑣 𝑗 ) = FALSE. This produces a satisfying truth assignment for the 3SAT instance 𝑉, 𝐶.

3.3.2

NP-Completeness Results for Bipartite Graphs

The NP-completeness results in the previous section apply to arbitrary graphs. The great majority of NP-completeness results, however, show that some computation problem remains NP-complete even when restricted to some special class of graphs. We illustrate this with the following theorem of Chang and Nemhauser [144] and three corollaries that can be derived from it. BIPARTITE DOMINATING SET

Instance: A bipartite graph 𝐺 = (𝑉, 𝐸) and a positive integer 𝑘. Question: Does 𝐺 have a dominating set of cardinality at most 𝑘? Theorem 3.4 ([144]) BIPARTITE DOMINATING SET is NP-complete, or equivalently, DOMINATING SET is NP-complete, even when restricted to bipartite graphs. Proof We first consider showing BIPARTITE DOMINATING SET ∈ NP. This part of the proof is quite straight forward. Given a bipartite graph 𝐺 and a set 𝑆 of vertices, one can easily check in polynomial time to see that |𝑆| ≤ 𝑘 and that 𝑆 is or is not a dominating set of 𝐺. Next we define a polynomial time reduction from DOMINATING SET for arbitrary graphs, which has already been shown to be NP-complete, to BIPARTITE DOMINATING SET. Given an instance of DOMINATING SET, that is, an arbitrary graph 𝐺 = (𝑉, 𝐸), with 𝑉 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 }, and a positive integer 𝑘, we construct an instance of BIPARTITE DOMINATING SET, that is, a bipartite graph 𝐺 ′ = (𝑉 ′ , 𝐸 ′ ) and a positive integer 𝑘 ′ = 𝑘 + 1, as follows. Let 𝑉 ′ = 𝑉 ∪ 𝑉 ′′ ∪ {𝑥, 𝑥 ′′ }, where 𝑉 ′′ = {𝑣 1′′ , 𝑣 2′′ , . . . , 𝑣 ′′ 𝑛 } is a copy of 𝑉, and 𝑥 and 𝑥 ′′ are two vertices not in 𝑉. Let 𝐸 ′ = {𝑢𝑣 ′′ , 𝑢 ′′ 𝑣 : 𝑢𝑣 ∈ 𝐸 } ∪ {𝑣𝑣 ′′ : 𝑣 ∈ 𝑉 } ∪ {𝑢𝑥 : 𝑢 ∈ 𝑉 } ∪ {𝑥𝑥 ′′ }. Finally, for any set 𝑆 ⊆ 𝑉 as part of an instance of DOMINATING SET, let 𝑆 ′ = 𝑆 ∪ {𝑥} be the corresponding set in the instance of BIPARTITE DOMINATING SET. For example, see Figure 3.2. We must show that (i) 𝐺, 𝑆 is a ‘yes’ instance of DOMINATING SET if and only if the 𝐺 ′ , 𝑆 ′ is a ‘yes’ instance of BIPARTITE DOMINATING SET. We must also show that (ii) for any instance 𝐺, 𝑘 of DOMINATING SET, we can construct the instance 𝐺 ′ , 𝑘 + 1 in polynomial time. But this is easy to see. The size of an instance of DOMINATING SET is O (𝑛 + 𝑚), where |𝑉 | = 𝑛 and |𝐸 | = 𝑚. The size of the constructed instance of BIPARTITE DOMINATING SET is 2𝑛 + 2 vertices and

Section 3.3. NP-Completeness of Domination Parameters

𝑎

57

𝑎 ′′

𝑎

𝑏 ′′

𝑏

𝑐′′

𝑐

𝑥 𝑥 ′′

𝑒

𝑏 𝑑 ′′ 𝑑 𝑐

𝑑

𝑒 ′′

(a) 𝐺

𝑒 (b) 𝐺 ′′

Figure 3.2 DOMINATING SET to BIPARTITE DOMINATING SET

2𝑛 + 2𝑚 + 1 edges. Thus, the size of an instance of BIPARTITE DOMINATING SET is at most a constant times the size of an instance of DOMINATING SET. This brings us to (i). Assume that an instance 𝐺, 𝑆 is a ‘yes’ instance of DOMINATING SET. We construct a set 𝑆 ′ = 𝑆 ∪ {𝑥} of the bipartite graph 𝐺 ′ . It is easy to see that if 𝑆 is a dominating set of 𝐺 of cardinality at most 𝑘, then 𝑆 ′ is a dominating set of 𝐺 ′ of cardinality at most 𝑘 + 1. Conversely, assume that 𝐺 ′ , 𝑆 ′ is a ‘yes’ instance of BIPARTITE DOMINATING SET. Since 𝑆 ′ is a dominating set of 𝐺 ′ , it must contain either vertex 𝑥 or vertex 𝑥 ′′ . We can assume, without loss of generality, that 𝑆 ′ contains 𝑥. Then 𝑥 dominates 𝑉 in 𝐺 ′ , and the remaining vertices in 𝑆 ′ dominate 𝑉 ′′ . It is easy to see that if there is some vertex 𝑣 ′′ ∈ 𝑆 ′ , then we can replace it with its corresponding vertex 𝑣 ∈ 𝑉 and still have a dominating set of 𝐺 ′ . Thus, we can make sure that there is a subset of vertices only in 𝑉 that dominates 𝑉 ′′ . This means that the same subset of vertices is a dominating set of 𝐺. Thus, a ‘yes’ instance of BIPARTITE DOMINATING SET corresponds to a ‘yes’ instance of DOMINATING SET. Notice that the dominating set of the bipartite graph 𝐺 ′ in the above proof is also a TD-set. From this we get an immediate corollary, using virtually the same proof. BIPARTITE TOTAL DOMINATING SET

Instance: A bipartite graph 𝐺 = (𝑉, 𝐸) and a positive integer 𝑘. Question: Does 𝐺 have a TD-set of cardinality at most 𝑘? Corollary 3.5 BIPARTITE TOTAL DOMINATING SET is NP-complete, or equivalently, TOTAL DOMINATING SET is NP-complete, even when restricted to bipartite graphs. We have thus demonstrated the NP-completeness of the decision problems associated with DOMINATING SET, INDEPENDENT DOMINATING SET, and

58

Chapter 3. Complexity and Algorithms for Domination in Graphs

TOTAL DOMINATING SET. In the next section, we review a much wider collection of NP-completeness results for DOMINATING SET when restricted to many other

classes of graphs.

3.3.3

Summary of Complexity Results for Graph Families

In 1990 Corneil and Stewart [201] provided a summary of results on the complexity of the DOMINATING SET decision problem for some 35 classes of graphs, where, for example, trees [P] ⊂ chordal bipartite graphs [NPc] indicates that the class of trees is a proper subclass of the class of chordal bipartite graphs, and that the domination number can be computed in polynomial time [P] for trees, but for chordal bipartite graphs the problem is NP-complete [NPc]. Subset inclusions like this show in some sense the boundary between classes of graphs for which DOMINATING SET can be solved in polynomial time and a larger class of graphs for which the problem is NP-complete. • trees ⊂ 𝑘-trees, fixed 𝑘 ⊂ 𝑘-trees, arbitrary 𝑘 ⊂ chordal graphs • trees ⊂ directed-path graphs ⊂ undirected path graphs ⊂ chordal graphs • split graphs ⊂ chordal graphs ⊂ weakly chordal graphs • interval graphs ⊂ directed-path graphs ⊂ strongly chordal graphs ⊂ chordal graphs • cographs ⊂ permutation graphs ⊂ comparability graphs ⊂ perfectly orderable graphs • permutation graphs ⊂ cocomparibility ⊂ asteroidal triple-free • cographs ⊂ 𝑃4 -reducible graphs ⊂ permutation graphs ⊂ trapezoid graphs ⊂ cocomparability graphs • trees ⊂ chordal bipartite graphs ⊂ bipartite graphs ⊂ comparability graphs • trees ⊂ generalized series-parallel graphs ⊂ partial 2-trees ⊂ partial 𝑘-trees, fixed 𝑘 ⊂ 𝑘-trees, arbitrary 𝑘 We summarize the complexity results for domination, total domination, and independent domination of some graph families in the following table. Using subset inclusions, other complexity results can be deduced. For example, chordal graphs ⊂ weakly chordal graphs, and since chordal graphs are in the class NPc for DOMINATING SET, weakly chordal graphs are also in NPc. For each graph family, the complexity P or NPc for the decision problems associated with domination, total domination, and independent domination is listed under 𝛾(𝐺), 𝛾t (𝐺), and 𝑖(𝐺), respectively in Table 3.1. Citations follow the complexity result.

3.4

A Representative Sample of Domination Algorithms for Trees

In this section, we present linear algorithms for computing the domination number, independent domination number, and total domination number of an arbitrary tree.

Section 3.4. A Representative Sample of Domination Algorithms for Trees Graph Class

𝛾(𝐺)

general graphs bipartite graphs comparability graphs chordal graphs split graphs 𝑘-trees, arbitrary 𝑘 undirected path graphs chordal bipartite graphs trees permutation graphs 𝑘-trees, fixed 𝑘 directed path graphs cographs cocomparability graphs interval graphs strongly chordal graphs partial 𝑘-trees, fixed 𝑘 asteroidal triple-free graphs series-parallel graphs circular-arc graphs

NPc [325, 510] NPc [144, 233] NPc [233] NPc [87] NPc [71, 200] NPc [199] NPc [87] NPc [601] P [187] P [100, 270] P [199] P [87] P [200] P [545] P [87, 269] P [269] P [38] P [544] P [523] P [147]

𝛾t (𝐺)

59

𝑖(𝐺)

NPc [510, 634] NPc [634] NPc [634] NPc [554, 555] NPc [554]

NPc [325, 510] NPc [200] NPc [200] P [268] P [268] P [268] NPc [555] P [268] P [211] NPc [211] P [555] P [74] P [100, 201, 546] P [270] P [38] P [38] P [268] P [200] P [270] P [545] P [545] P [72, 519] P [268, 269] P [142] P [268, 269] P [37] P [37] P [544] P [122] P [635] P [635] P [147] P [147]

Table 3.1 NP-completeness results for some graph families

3.4.1

Minimum Dominating Set

The first polynomial algorithm for computing the domination number of any nontrivial class of graphs, in this case trees, was published in 1975 by Cockayne et al. [187]. Their algorithm actually solves a more general version of the domination problem. Assume that the vertices of a graph 𝐺 = (𝑉, 𝐸) are partitioned into three sets, 𝐹, 𝐵, and 𝑅, where vertices in 𝐹 are called free, vertices in 𝐵 are called bound, and vertices in 𝑅 are called required. In the optional domination problem, given a graph 𝐺 and a partition {𝐹, 𝐵, 𝑅} of 𝑉, one must find a set 𝑆 ⊆ 𝑉 of minimum cardinality satisfying the following three conditions: (i) Every vertex in 𝐵 must either be in 𝑆 or adjacent to (dominated by) a vertex in 𝑆. (ii) Every vertex in 𝑅 must be in 𝑆. (iii) Vertices in 𝐹 need not be in 𝑆 nor adjacent to a vertex in 𝑆, but can be in 𝑆 in order to dominate vertices in 𝐵. Thus, (i) bound vertices must be dominated, either by being in 𝑆 or adjacent to a vertex in 𝑆, (ii) required vertices must be in the set 𝑆, and (iii) free vertices need not be dominated. A solution 𝑆 is called an optional dominating set. Therefore, if you are given an arbitrary graph 𝐺 = (𝑉, 𝐸) and the partition {∅, 𝑉, ∅} of 𝑉, then a solution 𝑆 to the optional domination problem is a minimum dominating set of 𝐺.

60

Chapter 3. Complexity and Algorithms for Domination in Graphs

Let 𝛾opt (𝐺), the optional domination number, equal the minimum cardinality of an optional dominating set for the partition {𝐹, 𝐵, 𝑅} of 𝑉. The correctness of the following algorithm for finding an optional dominating set of any tree is based on the following theorem. Theorem 3.6 ([187]) If 𝑇 = (𝑉, 𝐸) is an arbitrary tree, {𝐹, 𝐵, 𝑅} is a partiton of 𝑉, and 𝑣 is a leaf of 𝑇 whose neighbor is vertex 𝑢, then the following properties hold: (a) If 𝑣 ∈ 𝐹, then 𝛾opt (𝑇) = 𝛾opt (𝑇 − 𝑣). (b) If 𝑣 ∈ 𝐵 and 𝑇 ′ is the tree which results from deleting 𝑣 and placing 𝑢 in 𝑅, then 𝛾opt (𝑇) = 𝛾opt (𝑇 ′ ). (c) If 𝑢, 𝑣 ∈ 𝑅, then 𝛾opt (𝑇) = 1 + 𝛾opt (𝑇 − 𝑣). (d) If 𝑣 ∈ 𝑅 and 𝑢 ∉ 𝑅, and if 𝑇 ′ is the tree which results from deleting 𝑣 and placing 𝑢 in 𝐹, then 𝛾opt (𝑇) = 1 + 𝛾opt (𝑇 ′ ). Proof (a) Suppose that 𝑣 ∈ 𝐹. Since 𝑣 is free and need not be dominated, any optional dominating set of 𝑇 − 𝑣 is also an optional dominating set of 𝑇. Thus, 𝛾opt (𝑇) ≤ 𝛾opt (𝑇 − 𝑣).
Conversely, let 𝑆 be an optional dominating set of 𝑇. If 𝑣 ∈ 𝑆, then 𝑆 \ {𝑣} ∪ {𝑢} is an optional dominating set of 𝑇 − 𝑣, in which case 𝛾opt (𝑇 − 𝑣) ≤ 𝛾opt (𝑇). If 𝑣 ∉ 𝑆, then 𝑆 is also an optional dominating set of 𝑇 − 𝑣. Thus, 𝛾opt (𝑇 − 𝑣) ≤ 𝛾opt (𝑇). Thus, from both cases we can conclude that 𝛾opt (𝑇) = 𝛾opt (𝑇 − 𝑣). (b) Suppose that 𝑣 ∈ 𝐵. Let 𝑇 ′ be the tree which results from deleting 𝑣 and placing 𝑢 in 𝑅. Then any optional dominating set 𝑆 ′ of 𝑇 ′ is automatically an optional dominating set of 𝑇. Hence, 𝛾opt (𝑇) ≤ 𝛾opt (𝑇 ′ ). Conversely, let 𝑆 be an optional
dominating set of 𝑇. Since 𝑣 ∈ 𝐵, either 𝑣 ∈ 𝑆 or 𝑢 ∈ 𝑆. In either case, 𝑆 − {𝑣} ∪ {𝑢} is an optional dominating set of 𝑇 ′ . Hence, 𝛾opt (𝑇 ′ ) ≤ 𝛾opt (𝑇), and therefore, 𝛾opt (𝑇) = 𝛾opt (𝑇 ′ ). (c) Suppose that 𝑢, 𝑣 ∈ 𝑅. Every optional dominating set 𝑆 of 𝑇 must therefore contain both 𝑢 and 𝑣. It follows that 𝑆 \ {𝑣} is an optional dominating set of 𝑇 − 𝑣. Thus, 𝛾opt (𝑇 − 𝑣) ≤ |𝑆| − 1 = 𝛾opt (𝑇) − 1. Conversely, if 𝑆 ′ is an optional dominating set of 𝑇 − 𝑣, then since it must contain the vertex 𝑢, the set 𝑆 ′ ∪ {𝑣} is an optional dominating set of 𝑇. Thus, 𝛾opt (𝑇) ≤ 1 + |𝑆 ′ | = 1 + 𝛾opt (𝑇 − 𝑣). Consequently, 𝛾opt (𝑇) = 1 + 𝛾opt (𝑇 − 𝑣). (d) Suppose that 𝑣 ∈ 𝑅 and 𝑢 ∉ 𝑅. Let 𝑇 ′ be the tree which results from deleting 𝑣 and placing 𝑢 in 𝐹. Let 𝑆 ′ be an optional dominating set of 𝑇 ′ . Then clearly, 𝑆 ′ ∪ {𝑣} is an optional dominating set of 𝑇. Hence, 𝛾opt (𝑇) ≤ 1 + |𝑆 ′ | = 1 + 𝛾opt (𝑇 ′ ). Conversely, let 𝑆 be an optional dominating set of 𝑇, where by assumption, 𝑣 ∈ 𝑆 and 𝑢 ∈ 𝐹. It follows therefore that 𝑆 ′ = 𝑆 \ {𝑣} is an optional dominating set of 𝑇 − 𝑣, and therefore 𝛾opt (𝑇 ′ ) ≤ 𝛾opt (𝑇) − 1. Consequently, 𝛾opt (𝑇) = 1 + 𝛾opt (𝑇 ′ ). The four conditions in Theorem 3.6 suggest the following algorithm for computing the optional domination number of any tree 𝑇 with a given vertex partition {𝐹, 𝐵, 𝑅}, and hence for computing the domination number of any tree. The algorithm consists of a series of guarded commands, as originally introduced by Dijkstra, consisting of a guard 𝐺 𝑖 followed by a command 𝐴𝑖 , of the following form:

Section 3.4. A Representative Sample of Domination Algorithms for Trees

61

do 𝐺 1 : 𝐴1 𝐺 2 : 𝐴2 .. . 𝐺 𝑛 : 𝐴𝑛 od Each guard 𝐺 𝑖 is a Boolean expression, and each 𝐴𝑖 consists of a sequence of statements or actions to be executed. An algorithm using this form is executed as follows: find any guard 𝐺 𝑖 whose Boolean expression is TRUE and then execute the statements in 𝐴𝑖 . Repeat this process until none of the guards 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑛 are true. At any given point in the execution of an algorithm more than one guard may be true. In a deterministic algorithm you normally find the first guard 𝐺 𝑖 whose expression evaluates to TRUE and then execute the corresponding statements in 𝐴𝑖 . If several guards evaluate to TRUE and it does not matter which guard’s actions are executed, then the algorithm is called nondeterministic. The algorithm we present is nondeterministic.

Algorithm 1 Minimum Dominating Set Input : A tree 𝑇 = (𝑉, 𝐸) with each vertex labeled free, bound, or required Output : A minimum cardinality optional dominating set 𝑆 1 2

3 4

5

6

7

[Initialize] Set 𝑆 ← ∅ while 𝑇 has two or more vertices, execute any one of the following guarded commands do 𝐺 1 : 𝑇 has a free leaf 𝑣: set 𝑇 ← 𝑇 − 𝑣 𝐺 2 : 𝑇 has a bound leaf 𝑣 adjacent to a vertex 𝑢: label 𝑢 required; set 𝑇 ← 𝑇 − 𝑣 𝐺 3 : 𝑇 has a required leaf 𝑣 adjacent to a vertex 𝑢: set 𝑆 ← 𝑆 ∪ {𝑣}; if vertex 𝑢 is bound, then label 𝑢 free; set 𝑇 ← 𝑇 − 𝑣 od [The last vertex] if the one remaining vertex 𝑣 is bound or required, then set 𝑆 ← 𝑆 ∪ {𝑣}

We again point out, that while Algorithm 1 computes the minimum cardinality of an optional dominating set for a given vertex partition {𝐹, 𝐵, 𝑅}, it will compute the domination number 𝛾(𝑇) of any tree 𝑇 = (𝑉, 𝐸), given the partition {∅, 𝑉, ∅} of 𝑉, in which all vertices are initially labeled bound.

62

Chapter 3. Complexity and Algorithms for Domination in Graphs

3.4.2 Minimum Independent Dominating Set The following algorithm for computing the independent domination number 𝑖(𝑇) of any tree 𝑇 was constructed by Beyer et al. [74] in 1977. Without loss of generality, this algorithm assumes that all trees 𝑇𝑟 are rooted at some vertex 𝑟, and for every vertex 𝑣 ∈ 𝑉 (𝑇𝑟 ), we consider the subtree 𝑇𝑣 of 𝑇𝑟 rooted at 𝑣. Thus, for any vertex 𝑣, consider its parent vertex 𝑢, the edge 𝑢𝑣 in 𝑇𝑟 , and the subtree 𝑇𝑢 − 𝑉 (𝑇𝑣 ) consisting of all descendants of 𝑢 except for the descendants of 𝑢 in 𝑇𝑣 . We then consider what happens to independent dominating sets in both 𝑇𝑣 and 𝑇𝑢 − 𝑉 (𝑇𝑣 ) and how these sets can be merged to form independent dominating sets in the merged subtree 𝑇𝑢 . Recall that an 𝑖-set of a graph 𝐺 is a minimum independent dominating set in 𝐺. Consider any 𝑖-set 𝑆 in a rooted tree 𝑇𝑟 and how 𝑆 intersects the vertices in 𝑇𝑣 for any vertex 𝑣 ∈ 𝑉 (𝑇𝑟 ). Three possibilities exist: 1. 𝑣 ∈ 𝑆, 2. 𝑣 ∉ 𝑆 but 𝑣 is dominated by an immediate descendant (neighbor) of 𝑣 in 𝑇𝑣 , 3. 𝑣 ∉ 𝑆, no neighbor of 𝑣 in 𝑇𝑣 is in 𝑆 but the parent 𝑢 of 𝑣 is in 𝑆. Given this, we can define the following three numbers: IN(𝑣): the minimum cardinality of an independent set 𝑆 ⊂ 𝑇𝑣 which dominates all vertices in 𝑇𝑣 and for which 𝑣 ∈ 𝑆, OUTC(𝑣): the minimum cardinality of an independent set 𝑆 ⊂ 𝑇𝑣 which dominates all vertices in 𝑇𝑣 but which does not contain vertex 𝑣, OUTN(𝑣): the minimum cardinality of an independent set 𝑆 ⊂ 𝑇𝑣 which dominates all vertices in 𝑇𝑣 except vertex 𝑣. For these kinds of sets 𝑆, the parent 𝑢 of 𝑣 will be needed in order to dominate 𝑣. A linear algorithm for computing 𝑖(𝑇) for any tree 𝑇 can be constructed based on the following theorem. Theorem 3.7 ([74]) If 𝑇𝑢 and 𝑇𝑣 are two vertex-disjoint trees, rooted at vertices 𝑢 and 𝑣, respectively, and 𝑇𝑥 is the rooted tree obtained from 𝑇𝑢 and 𝑇𝑣 by adding the edge 𝑢𝑣 and rooting the merged tree at vertex 𝑢 but relabeling this new root 𝑥, then the following hold:  (a) IN(𝑥) = IN(𝑢) + min OUTC(𝑣), OUTN(𝑣) .  (b) OUTC(𝑥) = min OUTN(𝑢) + IN(𝑣), OUTC(𝑢) + OUTC(𝑣), OUTC(𝑢) + IN(𝑣) . (c) OUTN(𝑥) = OUTN(𝑢) + OUTC(𝑣). Proof (a) Let 𝑆 be an 𝑖-set of a rooted tree 𝑇𝑥 by merging two rooted trees 𝑇𝑢 and 𝑇𝑣 , and assume that 𝑢 = 𝑥 ∈ 𝑆. Let 𝑆𝑢 = 𝑆 ∩ 𝑇𝑢 and 𝑆 𝑣 = 𝑆 ∩ 𝑇𝑣 . Clearly, 𝑢 ∈ 𝑆𝑢 and |𝑆𝑢 | = |IN(𝑢)|, else 𝑆 does not have minimum cardinality.  Also, if 𝑢 ∈ 𝑆, then 𝑣 ∉ 𝑆, else 𝑆 is not an independent set. Therefore, |𝑆 𝑣 | = min OUTC(𝑣), OUTN(𝑣) . (b) Let 𝑆𝑢 and 𝑆 𝑣 be as defined above in (a), except that we assume 𝑢 ∉ 𝑆. Vertex 𝑥 = 𝑢 must be dominated in one of three ways, namely: (i) by vertex 𝑣, (ii) by a child of 𝑢 in 𝑇𝑢 , or (iii) by both 𝑣 and a child of 𝑢 in 𝑇𝑢 . In case (i), it follows that |𝑆| = IN(𝑣) + OUTN(𝑢). In case (ii), it follows that |𝑆| = OUTC(𝑢) + OUTC(𝑣). In case (iii), it follows that |𝑆| = OUTC(𝑢) + IN(𝑣).

Section 3.4. A Representative Sample of Domination Algorithms for Trees

63

(c) Let 𝑆 be an 𝑖-set in 𝑇𝑥 that does not contain 𝑥 and such that 𝑥 is the only vertex in 𝑇𝑥 that is not dominated. In this case, it follows that |𝑆| = OUTN(𝑢) + OUTC(𝑣).  Corollary 3.8 If 𝑢 is the root of a tree 𝑇, then 𝑖(𝑇) = min IN(𝑢), OUTC(𝑢) . A simple algorithm for computing 𝑖(𝑇) for any tree 𝑇 can be constructed as follows. Arbitrarily root the tree 𝑇 of order 𝑛 at any vertex 𝑟. Order the vertices according to a breadth-first ordering 𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 , where 𝑣 1 = 𝑟. Create an array parent[1, 2, . . . , 𝑛], where for every 2 ≤ 𝑗 ≤ 𝑛, parent[ 𝑗] = 𝑖 for some 𝑖 < 𝑗 and 𝑣 𝑖 is the parent of vertex 𝑣 𝑗 in the breadth-first ordering.

Algorithm 2 Minimum Independent Dominating Set Input : A tree 𝑇 = (𝑉, 𝐸) given by an array parent[1, 2, . . . , 𝑛] Output : |𝑆| = 𝑖(𝑇) 1 2 3 4 5

6 7 8 9

10 11

12

[Initialize the values of IN(𝑣), OUTC(𝑣), OUTN(𝑣)] for 1 ≤ 𝑣 ≤ 𝑛 do IN(𝑣) = 1 OUTC(𝑣) = 𝑛 OUTN(𝑣) = 0 od [Propagate values from vertex 𝑣 𝑛 to 𝑣 2 ] for 𝑣 = 𝑛 down to 2 do Let 𝑢 = parent[𝑣]  IN(𝑢) = IN(𝑢) + min OUTC(𝑣), OUTN(𝑣)  OUTC(𝑢) = min OUTN(𝑢) + IN(𝑣), OUTC(𝑢) + OUTC(𝑣), OUTC(𝑢) + IN(𝑣) OUTN(𝑢) = OUTN(𝑢) + OUTC(𝑣) od [Root vertex 𝑟 = 1]  |𝑆| = min IN(1), OUTC(1)

3.4.3

Minimum Total Dominating Set

The following algorithm, due to Laskar et al. [555] in 1984 for computing the total domination number of an arbitrary nontrivial tree, actually solves a more general problem, similar to Algorithm 1 presented earlier in this section. In an optional total dominating set vertices can have one of four different labels: (a) free, meaning that the vertex does not need to be in the TD-set; this happens when the vertex has already been totally dominated by a vertex in 𝑆, (b) bound, meaning that the vertex is neither in 𝑆 nor adjacent to a vertex in 𝑆, but must be totally dominated,

64

Chapter 3. Complexity and Algorithms for Domination in Graphs

(c) required-1, meaning it is in 𝑆 to dominate a vertex in 𝑉, but it is not yet adjacent to a vertex in 𝑆, or (d) required-2, meaning it is a vertex in 𝑆 and is also adjacent to a vertex in 𝑆. Recall that a 𝛾t -set of a graph 𝐺 is a minimum TD-set in 𝐺. In order for the algorithm to find a 𝛾t -set in a tree, all vertices are initially labeled bound, meaning that they must be either in the TD-set 𝑆 being computed or dominated by a vertex in 𝑆. The algorithm proceeds by processing only one leaf 𝑣 in the tree at a time, whose only neighbor is vertex 𝑢. After the vertex 𝑣 has been processed, it is deleted from the tree, resulting in a smaller tree.

Algorithm 3 Minimum Total Dominating Set Input : A tree 𝑇 = (𝑉, 𝐸) with all vertices labeled bound Output : A 𝛾t -set 𝑆 1 2

3 4

5

6

7

8

9

10 11 12 13 14 15 16 17 18 19

[Initialize] Set 𝑆 ← ∅ while 𝑇 has three or more vertices, execute any one of the following guarded commands do 𝐺 1 : 𝑇 has a free leaf 𝑣: set 𝑇 ← 𝑇 − 𝑣 𝐺 2 : 𝑇 has a bound leaf 𝑣 adjacent to a free or required-2 vertex 𝑢: label 𝑢 required-2; set 𝑇 ← 𝑇 − 𝑣 𝐺 3 : 𝑇 has a bound leaf 𝑣 adjacent to a bound or required-1 vertex 𝑢: label 𝑢 required-1; set 𝑇 ← 𝑇 − 𝑣 𝐺 4 : 𝑇 has a required-1 leaf 𝑣 adjacent to a vertex 𝑢: set 𝑆 ← 𝑆 ∪ {𝑣}; label 𝑢 required-2; set 𝑇 ← 𝑇 − 𝑣 𝐺 5 : 𝑇 has a required-2 leaf 𝑣 adjacent to a required-1 or required-2 vertex 𝑢: set 𝑆 ← 𝑆 ∪ {𝑣}; label 𝑢 required-2; set 𝑇 ← 𝑇 − 𝑣 𝐺 6 : 𝑇 has a required-2 leaf 𝑣 adjacent to a free or bound vertex 𝑢: set 𝑆 ← 𝑆 ∪ {𝑣}; label 𝑢 free; set 𝑇 ← 𝑇 − 𝑣 od [Two remaining vertices 𝑢 and leaf 𝑣] case (label(𝑢), label(𝑣)) of :𝑆←𝑆 (free, free) (bound, free) : 𝑆 ← 𝑆 ∪ {𝑣} (bound, bound) : 𝑆 ← 𝑆 ∪ {𝑢, 𝑣} (required-1, . . . ) : 𝑆 ← 𝑆 ∪ {𝑢, 𝑣} (. . . , required-1) : 𝑆 ← 𝑆 ∪ {𝑢, 𝑣} (required-2, free) : 𝑆 ← 𝑆 ∪ {𝑢} (required-2, bound) : 𝑆 ← 𝑆 ∪ {𝑢} (required-2, required-2) : 𝑆 ← 𝑆 ∪ {𝑢, 𝑣} endcase

Section 3.5. Early Domination Algorithms and NP-Completeness Results

65

The correctness of Algorithm 3 can be seen by the entries in Table 3.2. Consider vertex 𝑣 to be the root of a subtree 𝑇𝑣 positioned to the right of vertex 𝑢, which is the root of its own subtree 𝑇𝑢 , with an edge joining 𝑣 to 𝑢. The label of vertex 𝑣 indicates that in the subtree 𝑇𝑣 we have found a 𝛾t -set 𝑆 𝑣 subject to one of four conditions: (a) vertex 𝑣, labeled free (column 2), is not in 𝑆 𝑣 but is dominated by a vertex in 𝑆 𝑣 , in which case the label of vertex 𝑢 need not change. (b) vertex 𝑣, labeled bound (column 3), is not dominated by any vertex in 𝑆 𝑣 but all other vertices in 𝑇𝑣 are dominated by 𝑆 𝑣 ; in this case vertex 𝑢 must be placed into 𝑆𝑢 in order to dominate vertex 𝑣, the label of 𝑢 then becoming required-2 if it already has a neighbor in 𝑆𝑢 and is currently labeled either free or required-2, or the label of 𝑢 becoming required-1, in case it has no neighbor in 𝑆𝑢 and its label is currently either bound or required-1. (c) vertex 𝑣, labeled required-1 (column 4), is in 𝑆 𝑣 but has no neighbor in 𝑆 𝑣 , which then requires vertex 𝑢 to be placed into 𝑆𝑢 and be labeled required-2, since it will be in 𝑆𝑢 with a neighbor in 𝑆 𝑣 (vertex 𝑣). (d) vertex 𝑣, labeled required-2 (column 5), is in 𝑆 𝑣 and also has a neighbor in 𝑆 𝑣 . In this case, vertex 𝑢 becomes dominated and can be labeled free or can be relabeled required-2 if it is currently labeled required-1. label(𝑣) label(𝑢)

free

bound

required-1

required-2

free bound required-1 required-2

free bound required-1 required-2

required-2 required-1 required-1 required-2

required-2 required-2 required-2 required-2

free free required-2 required-2

Table 3.2 Delete 𝑇𝑣 and label 𝑢 as indicated in the table

3.5

Early Domination Algorithms and NP-Completeness Results

In this section, we present a chronology of early domination algorithms and NPcompleteness results. When a parameter is referred to as weighted, the vertices are labeled with nonnegative integers, and one seeks a set 𝑆 for the associated parameter for which the sum of the weights of the vertices in 𝑆 is a minimum or a maximum, depending on the parameter. 1959

• Maghout [580] presented exponential-time algorithms for the independence number 𝛼(𝐷) and in-domination number 𝛾in (𝐷) in digraphs.

Chapter 3. Complexity and Algorithms for Domination in Graphs

66 1966

• Daykin and Ng [217] presented linear algorithms for unweighted and weighted independence and domination in directed trees; unfortunately, both of the unweighted algorithms for independence and domination contain errors, as do the attempted proofs of correctness of these algorithms. They are noteworthy, nevertheless, for being in some sense ahead of their time, for much was to follow this pioneering paper. 1972

• Gavril [330] presented an O (𝑛3 ) algorithm for independence in chordal graphs, that is, graphs in which every cycle of length greater than three has a chord. • Nieminen [612] presented a greedy algorithm for finding a minimal, but not necessarily minimum, dominating set in an arbitrary graph. 1973

• Gavril [331] presented an O (𝑛3 ) algorithm for independence in circle graphs, that is, graphs whose vertex set is a family of chords of a given circle and in which two vertices are joined by an edge if and only if the corresponding chords intersect. 1974

• Gavril [332] presented an O (𝑛3 ) algorithm for independence in circular arc graphs, that is, graphs whose vertex set is a family of arcs of a given circle and in which two vertices are joined by an edge if and only if the corresponding arcs intersect. 1975

• Cockayne et al. [187] presented the first domination algorithm in graphs, a linear algorithm for domination in trees. • Mitchell et al. [595] presented linear algorithms for trees for independence, vertex covering, and matching. 1977

• Beyer et al. [74] presented a linear algorithm for independent domination in trees. • In her PhD thesis, Mitchell [592, 594] presented a linear algorithm for edge domination 𝛾 ′ (𝑇) in trees, or domination in line graphs of trees, which showed that for trees, 𝛾 ′ (𝑇) = 𝑖 ′ (𝑇), or equivalently that 𝛾(𝐿(𝑇)) = 𝑖(𝐿(𝑇)). She also presented linear algorithms for independence in 𝑘-trees, including trees and maximal outerplanar graphs. 1978

• Natarajan and White [607] presented a linear algorithm for weighted domination in trees. 1979

• In their book [325], Garey and Johnson listed many NP-completeness and algorithmic results for domination parameters. In particular, they stated that INDEPENDENT DOMINATING SET is NP-complete. They also stated that DOMINATING SET is NP-complete for general graphs, planar graphs with Δ(𝐺) ≤ 3, 4-regular

Section 3.5. Early Domination Algorithms and NP-Completeness Results

67

planar graphs, line graphs of planar graphs with Δ(𝐺) ≤ 3, and line graphs of bipartite graphs with Δ(𝐺) ≤ 3. • Mitchell et al. [593] presented linear algorithms on trees for independence, matching, and domination, using recursive representations, or parent arrays, of trees. 1980

• Yannakakis and Gavril [760] constructed an NP-completeness proof for domination and independent domination in line graphs of graphs, or equivalently, an NP-completeness proof for edge domination. 1981

• Dewdney [233] constructed NP-completeness proofs for domination in bipartite graphs and comparability graphs. • Farber [267] presented a linear algorithm for weighted domination in trees. 1982

• Booth and Johnson [87] constructed NP-completeness proofs for domination in chordal graphs and undirected path graphs. They also presented a linear algorithm for domination in directed path graphs, given an appropriate path representation. • Chang [141] in private communication claimed to have constructed an NPcompleteness proof for integer weighted independent domination in chordal graphs. • Chang and Nemhauser [144] constructed an NP-completeness proof for domination in bipartite graphs. • Farber [268] presented a polynomial time algorithm for weighted independent domination in chordal graphs, and thus, in strongly chordal graphs and interval graphs. • Takamizawa et al. [700] presented a ground-breaking, linear algorithm for independence in two-terminal, series-parallel graphs. This would lead to several other series-parallel algorithms. 1983

• Kikuno et al. [523] presented the first domination algorithm in series-parallel graphs, whose running time is linear. • Laskar and Pfaff [554] constructed an NP-completeness proof for total domination in split graphs, and hence, in chordal graphs. • Pfaff et al. [634] constructed an NP-completeness proof for total domination in bipartite graphs. 1984

• Bertossi [71] constructed NP-completeness proofs for domination in split graphs (a subclass of chordal graphs) and in bipartite graphs. • Corneil and Perl [200] constructed NP-completeness proofs for independent domination in bipartite graphs and in comparability graphs. • Farber [269] presented polynomial algorithms for weighted domination and weighted independent domination in strongly chordal graphs.

Chapter 3. Complexity and Algorithms for Domination in Graphs

68

• Pfaff et al. [635] presented linear algorithms for independent domination and total domination in series-parallel graphs. • Laskar et al. [555] presented a linear algorithm for total domination in trees. They also constructed an NP-completeness proof for total domination in undirected path graphs, that is, intersection graphs of families of paths in a graph. 1985


• Chang and Nemhauser [144] presented O |𝑉 ||𝐸 | algorithms for domination and independence in what they call nearly chordal graphs, a family of graphs which properly contain all chordal graphs. These are graphs which are too complex to define in detail here, but roughly they are graphs the maximum length of an induced cycle (or hole) in which is 4, but if there is a 4-hole, then two conditions on each 4-hole must hold. The authors also showed that for what are called odd sun-free chordal graphs, domination and independence can be solved in polynomial time. • Farber and Keil [270] presented O (𝑛3 ) algorithms for weighted domination and weighted independent domination in permutation graphs. They also presented an O (𝑛2 ) algorithm for domination in permutation graphs. • Johnson [511] constructed an NP-completeness proof for domination in arbitrary grid graphs, which include all subgraphs of complete grid graphs, 𝑃𝑚 □ 𝑃𝑛 , for arbitrary 𝑚, 𝑛 ≥ 1. In 2011 Goncalves et al. [362] presented 16 formulas for the domination numbers 𝛾(𝑃𝑚 □ 𝑃𝑛 ), for all 𝑚, 𝑛 ≥ 1. These formulas are given in Chapter 17. 1986

• Bertossi [72] presented an O (𝑛2 ) algorithm for total domination in interval graphs. • Hedetniemi et al. [450] presented a linear algorithm for domination in cacti. • Keil [519] presented an O (𝑛 + 𝑚) algorithm for total domination in interval graphs. 1987

• Brandstädt and Kratsch [100] presented an O (𝑛3 ) algorithm for weighted domination in permutation graphs, an O (𝑛 log2 𝑛) algorithm for independent domination in permutation graphs, and an O (𝑛2 ) algorithm for weighted independent domination in permutation graphs. • Corneil and Keil [199] presented a polynomial algorithm for domination in 𝑘-trees, for fixed 𝑘, but constructed an NP-completeness proof for domination in 𝑘-trees for arbitrary 𝑘. • Hare and Hedetniemi [388] presented a linear algorithm for domination in 𝑘 × 𝑛 knights graphs, for fixed 𝑘. • Hare et al. [390] presented a linear algorithm for upper domination in generalizedseries-parallel graphs, a class of graphs which properly contains two-terminal, series-parallel graphs, trees, outerplanar graphs, unicyclic graphs, 2-trees, and cacti.

Section 3.5. Early Domination Algorithms and NP-Completeness Results

69

• In his PhD thesis, Wimer [752] presented a linear algorithm for upper domination in partial 𝑘-chordal graphs. 1988

• Atallah et al. [40] presented an O (𝑛 log2 𝑛) algorithm for independent domination in permutation graphs. • Bertossi and Gori [73] presented an O (𝑛 log 𝑛) algorithm for total domination in weighted interval graphs. • Ramalingam and Rangan [645] presented linear algorithms for domination, independent domination, and total domination in interval graphs. 1989

• Arnborg and Proskurowski [38] presented linear algorithms for domination and independence in partial 𝑘-trees. 1990

• Cheston et al. [167] constructed the first NP-completeness proof for upper domination. • Corneil and Stewart [201] constructed NP-completeness proofs for domination and total domination in 𝑘-CUBs, for 𝑘 ≥ 2. They also presented polynomial algorithms for domination in 1-CUBs and total domination in permutation graphs. • Elmallah and Stewart [256] presented polynomial algorithms for domination, independent domination, and total domination in 𝑘-polygon graphs, that is, intersection graphs of straight line chords inside a convex 𝑘-gon, for 𝑘 ≥ 3. • Kratochvíl and Nešetřil [543] constructed NP-completeness proofs for independence in what they call 2-DIR and PURE-3-DIR graphs. These refer to intersection graphs of line segments in the plane which have at most 𝑘 different directions, and PURE refers to the condition that parallel segments on the same infinite line do not intersect. 1993

• Kratsch and Stewart [545] presented O (𝑛6 ), O (𝑛6 ), and O (𝑛3 ) algorithms, respectively, for domination, total domination, and independent domination in cocomparability graphs (complements of comparability graphs). • Keil [520] constructed NP-completeness proofs for domination and total domination in circle graphs. 1994

• Fellows et al. [288] constructed NP-completeness proofs for independence and upper domination in graphs with Δ(𝐺) ≤ 5, trestled graphs of index 𝑘 for any 𝑘 ≥ 1, planar graphs, and triangle-free graphs. • Liang and Rhee [563] presented linear O (𝑛 + 𝑚) algorithms for two independent set problems, including weighted independent set, in permutation graphs. 1997

• Chang [146] constructed NP-completeness proofs for weighted domination and total domination in co-bipartite graphs (complements of bipartite graphs). He

Chapter 3. Complexity and Algorithms for Domination in Graphs

70

also presented polynomial algorithms for bounded weighted domination and bounded weighted total domination in cocomparability graphs. 1998

• Fricke et al. [313] constructed NP-completeness proofs for independence in 𝑘-regular graphs, for 𝑘 ≥ 3, and for the upper domination number in 𝑘-regular graphs, for 𝑘 ≥ 6. • Chang [147] presented a unified approach to the design of algorithms for weighted domination, weighted independence, and total domination for both interval graphs (given an interval model with endpoints sorted) and
circular-arc graphs. These algorithms run in either O (𝑛) or O 𝑛 log(log 𝑛) time on interval graphs and O (𝑚 + 𝑛) time on circular-arc graphs. 1999

Since 1999 more than 1100 papers have been published on various kinds of domination algorithms and complexity in a wide variety of classes of graphs. In this section, we have reviewed only what might be considered the early foundational papers on algorithms and complexity of domination in graphs. Indeed, one could publish several books alone on this topic.

3.6

Other Sources for Domination Algorithms and Complexity

Other than the more than 1100 papers on domination algorithms and complexity that have been published since 1999, several books have been published which contain much of this material, which we describe in this section. The book Fundamentals of Domination in Graphs, by Haynes et al. [417], contains the following: • Section 1.11 An Introduction to NP-Completeness. • Section 1.12 NP-Completeness of the Domination Problem, containing an NPcompleteness proof of DOMINATING SET. • Chapter 12 Domination Complexity and Algorithms, 27 pages, contains NPcompleteness proofs of DOMINATING SET when restricted to bipartite graphs and when restricted to chordal graphs. It mentions, with references, published NPcompleteness results for lower irredundance, domination, independent domination, independence, upper domination, upper irredundance, connected domination, and total domination. Interestingly, no mention is made of upper total domination. But in her PhD thesis, McRae [588] settled the NP-completeness of upper total domination. Also presented in this chapter are polynomial algorithms for domination in trees, domination in interval graphs, total domination in interval graphs, independent domination in interval graphs, vertex independence in interval graphs, and minimum weight independent domination in permutation graphs. The book Domination in Graphs: Advanced Topics, edited by Haynes et al. [416], contains the following two chapters:

Section 3.6. Other Sources for Domination Algorithms and Complexity

71

• Chapter 8 Algorithms, by Kratsch, 41 pages and 155 references, contains algorithms for minimum weight dominating set in strongly chordal graphs, connected domination in cocomparability graphs, minimum weight total domination in interval graphs, independent domination in permutation graphs, and dominating clique in dually chordal graphs. • Chapter 9 Complexity Results, by Hedetniemi, McRae, and Parks, 37 pages and 34 references, contains NP-completeness results for bipartite graphs for independent domination, 2-maximal matchings, and minimum maximal strong matchings. In addition, it contains NP-completeness results for both bipartite graphs and chordal graphs for: perfect domination, efficient domination, efficient total domination, minimum maximal strong stable sets or 2-packings, domination, total domination, odd domination, weak vertex-edge domination, lower irredundance, closed open irredundance, open open irredundance, and open irredundance. In addition, in Part III of Structures of Domination in Graphs, edited by Haynes et al. [414], published in 2021, one can find the following chapters on domination algorithms and complexity: • Algorithms and Complexity of Signed, Minus, and Majority Domination, by Hedetniemi, McRae, and Mohan, with 30 pages and 36 references. • Algorithms and Complexity of Power Domination in Graphs, by Hedetniemi, McRae, and Mohan, with 24 pages and 43 references. • Self-Stabilizing Domination Algorithms, by Hedetniemi, with 36 pages and 72 references. • Algorithms and Complexity of Alliances in Graphs, by Hedetniemi, with 15 pages and 21 references. In addition to the above, the book Exact Exponential Algorithms, by Fomin and Kratsch [307], contains results on exact exponential algorithms for independence and domination in graphs. We quote here from Math Reviews MR3234973 about this book: Moderately exponential-time algorithm is the somewhat euphemistic alternative name for an area of algorithms that has taken off and flourished in the past decade. The main thrust is how to cope with problems that have been proven to be NP-hard, yet must be solved in practice, and not only approximately or merely heuristically, but exactly and reliably. Fedor Fomin and Dieter Kratsch have always been in the front line of these developments. They are therefore predestined to write the first textbook in this area. The book covers the main techniques that have been developed section by section: branching, dynamic programming, inclusion-exclusion, treewidth, measure & conquer, subset convolution, local search, split and list, and trade-offs like time versus space. It mostly sticks to only a few problems to illustrate the techniques: independence and domination in graphs, coloring, traveling salesman, and satisfiability. All statements directly concerning the algorithmic techniques come with proofs, only

72

Chapter 3. Complexity and Algorithms for Domination in Graphs some purely combinatorial yet very technical theorems are merely stated, with pointers to the literature in each such case. The coverage of the recent literature on the area is another strength of the book, which can be, overall, recommended as a textbook for a master (or PhD level) course on this still new and hot topic in algorithms.

It is also worth mentioning that the 1994 PhD thesis of McRae [588] contains 81 original NP-completeness results for bipartite graphs, chordal graphs, line graphs, and line graphs of bipartite graphs, for the decision problems relating to: domination, independent domination, total domination, perfect domination, efficient domination, efficient total domination, independence, upper domination, upper total domination, and upper irredundance. We conclude by simply listing by subject area a sampling of more recent papers on domination algorithms and complexity. For domination, see [260, 497, 544, 614, 627]. For independent domination, see [89, 216, 247, 499, 565, 624]. For upper domination, see [2, 62, 91]. For greedy algorithms, see [670, 671, 788]. For self-stabilizing domination algorithms, see [64, 168, 169, 234, 323, 346–348, 367, 441, 442, 449, 503, 690, 718, 756]. For approximation algorithms and complexity, see [170, 212, 213, 374, 611, 623]. For fixed parameter algorithms and complexity, see [13, 90, 244, 292, 371, 515, 584, 706]. For domination algorithms on Cartesian products, see [205, 207, 334]. For online domination algorithms, see [92, 375]. For exact exponential domination algorithms, see [169, 234, 306, 308, 323, 347, 348, 367, 441, 442, 448, 449, 503, 610, 669, 677, 690, 702, 718, 756]. For approximation algorithms and complexity, see [170, 212, 213, 374, 611, 623].

Chapter 4

General Bounds 4.1 Introduction As we have seen in Chapter 3, the decision problems associated with computing the domination, total domination, and independent domination numbers of arbitrary graphs are all NP-complete. Given the difficulty of determining the exact values of these domination numbers, much of the research involves the determination of tight lower and upper bounds for these numbers. In this chapter, we present some of the more basic bounds on the domination, total domination, and independent domination numbers of graphs. Additional bounds on these parameters will be given throughout the text, particularly in Chapter 6 in terms of minimum degree and Chapter 8 in terms of size. Recall that we use the abbreviation “TD-set” for a “total dominating set,” and “ID-set” for an “independent dominating set.” Further, recall that a dominating vertex in a graph 𝐺 of order 𝑛 is a vertex (of degree 𝑛 − 1) adjacent to every other vertex in 𝐺. The corona 𝐺 ◦ 𝐾1 of a graph 𝐺 is the graph obtained from 𝐺 by adding for each vertex 𝑣 ∈ 𝑉 a new vertex 𝑣 ′ and the edge 𝑣𝑣 ′ . A packing in a graph 𝐺 is a set of vertices whose closed neighborhoods are pairwise disjoint. Thus, if 𝑆 is a packing in 𝐺, then N[𝑢] ∩ N[𝑣] = ∅ for all 𝑢, 𝑣 ∈ 𝑆, implying that 𝑑 (𝑢, 𝑣) ≥ 3 for all 𝑢, 𝑣 ∈ 𝑆. The packing number 𝜌(𝐺) is the maximum cardinality of a packing in 𝐺. An open packing in a graph 𝐺 is a set of vertices whose open neighborhoods are pairwise disjoint. Thus, if 𝑆 is an open packing in 𝐺, then N(𝑢) ∩ N(𝑣) = ∅ for all 𝑢, 𝑣 ∈ 𝑆. The open packing number 𝜌 o (𝐺) is the maximum cardinality of an open packing in 𝐺. A perfect packing in 𝐺 is a packing whose closed neighborhoods partition 𝑉 (𝐺), and a perfect open packing in 𝐺 is an open packing whose open neighborhoods partition 𝑉 (𝐺). A perfect packing is also called an efficient dominating set and a perfect open packing is called an efficient total dominating set, to be discussed further in Chapter 9. A graph is claw-free if it contains no induced 𝐾1,3 . For ease of discussion, we say that an edge is between two sets 𝑋 and 𝑌 if it is incident to a vertex in 𝑋 and to a vertex in 𝑌 . © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_4

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Chapter 4. General Bounds

74

4.2 Domination and Maximum Degree In this section, we present some elementary bounds on the domination, total domination, and independent domination numbers of a graph in terms of the maximum degree. We first give some trivial properties of domination in graphs. Adding edges to a graph cannot increase its domination number. This yields the following observation. Observation 4.1 If 𝐻 is a spanning subgraph of a graph 𝐺, then 𝛾(𝐺) ≤ 𝛾(𝐻). Let 𝐻 be a spanning subgraph of a graph 𝐺, where 𝐻 is isolate-free. Thus, 𝑉 (𝐻) = 𝑉 (𝐺) and 𝐸 (𝐻) ⊆ 𝐸 (𝐺), and 𝛿(𝐻) ≥ 1. Let 𝑆 be a 𝛾t -set of 𝐻. Hence, every vertex not in 𝑆 has a neighbor in 𝑆 in the graph 𝐻 and the subgraph 𝐻 [𝑆] of 𝐻 induced by 𝑆 contains no isolated vertex. Adding edges to 𝐻 to rebuild the graph 𝐺 preserves these two properties, that is, every vertex not in 𝑆 has a neighbor in 𝑆 in the graph 𝐺 and the subgraph 𝐻 [𝑆] of 𝐻 induced by 𝑆 contains no isolated vertex. Thus, 𝑆 is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ |𝑆| = 𝛾t (𝐻). We state this trivial observation formally as follows. Observation 4.2 If 𝐻 is an isolate-free spanning subgraph of a graph 𝐺, then 𝛾t (𝐺) ≤ 𝛾t (𝐻).

4.2.1

Domination Number and Maximum Degree

In 1973 Berge [68] observed that if 𝑣 is a vertex of maximum degree in a graph 𝐺 = (𝑉, 𝐸) of order 𝑛, then the vertex 𝑣 together with all its non-neighbors forms a dominating set of 𝐺, and so 𝛾(𝐺) ≤ |𝑉 \ N(𝑣)| = 𝑛 − deg(𝑣) = 𝑛 − Δ(𝐺). If 𝐹 is a graph of order 𝑘 ≥ 1 that contains a dominating vertex, then the corona 𝐺 = 𝐹 ◦ 𝐾1 of 𝐹 is a graph of order 𝑛 = 2𝑘 with maximum degree Δ(𝐺) = 𝑘 and domination number 𝛾(𝐺) = 𝑘 = 𝑛 − Δ(𝐺). Thus, we have the following. Theorem 4.3 ([68]) If 𝐺 is a graph of order 𝑛, then 𝛾(𝐺) ≤ 𝑛 − Δ(𝐺), and this bound is tight. In 1979 Walikar et al. [741] observed that in a 𝛾-set 𝑆 of a graph 𝐺, every vertex of 𝑆 dominates itself and at most Δ(𝐺) other vertices. Thus, at most Δ(𝐺)
+ 1 vertices are dominated by each vertex of 𝑆, implying that at most Δ(𝐺) + 1 |𝑆| distinct every vertex
is dominated by the set 𝑆, vertices are dominated by the set 𝑆. Since
we therefore have that 𝑛 ≤ Δ(𝐺) + 1 |𝑆| = Δ(𝐺) + 1 𝛾(𝐺). Further, if there is equality in this inequality, then the following conditions hold: (i) the set 𝑆 is an independent set, (ii) no two vertices of 𝑆 have a common neighbor, and (iii)
each vertex of 𝑆 has degree Δ(𝐺). In particular, in this case when 𝑛 = Δ(𝐺) + 1 𝛾(𝐺), we have 𝛾(𝐺) = 𝑖(𝐺). These observations yield the following trivial lower bound on the domination number of a graph in terms of its order and maximum degree. Theorem 4.4 ([741]) If 𝐺 is a graph of order 𝑛, then 𝛾(𝐺) ≥

𝑛 , 1 + Δ(𝐺)

Section 4.2. Domination and Maximum Degree

75

with equality if and only if every 𝛾-set in 𝐺 is a perfect packing and every vertex in such a 𝛾-set has degree Δ(𝐺). As an application of Theorem 4.4, we can readily determine the domination and independent domination numberfor a path 𝑃𝑛 and a cycle 𝐶𝑛 of order 𝑛 ≥ 3. By Theorem 4.4, we have 𝛾(𝐶𝑛 ) ≥ 𝑛3 . Choosing every third vertex on the path 𝑃𝑛 , starting with the first vertex if 𝑛 ≡ 1 (mod 3) and starting with the second   vertex otherwise, produces an ID-set of the path 𝑃𝑛 , and so 𝛾(𝑃𝑛 ) ≤ 𝑖(𝑃𝑛 ) ≤ 𝑛3 . Since adding edges cannot increase   the domination number, 𝛾(𝐶𝑛 ) ≤ 𝛾(𝑃𝑛 ). Combining these inequalities, we have 𝑛3 ≤ 𝛾(𝐶𝑛 ) ≤ 𝛾(𝑃𝑛 ) ≤ 𝑖(𝑃𝑛 ) ≤ 𝑛3 , and hence equality     must hold throughout this inequality chain. Moreover, 𝑛3 = 𝛾(𝐶𝑛 ) ≤ 𝑖(𝐶𝑛 ) ≤ 𝑛3 , once again yielding equality throughout the inequality chain. We state this formally as follows. Proposition 4.5 For 𝑛 ≥ 3, 𝛾(𝐶𝑛 ) = 𝛾(𝑃𝑛 ) = 𝑖(𝐶𝑛 ) = 𝑖(𝑃𝑛 ) =

𝑛

.

3

We state next several other upper bounds on the domination number of a graph in terms of its order and both minimum and maximum degrees. The following bound was determined in 1986 by Marcu [583]. Theorem 4.6 ([583]) If 𝐺 is a graph of order 𝑛, then

  𝑛 − Δ(𝐺) − 1 𝑛 − 𝛿(𝐺) − 2 𝛾(𝐺) ≤ + 2. 𝑛−1 Flach and Volkmann [305] in 1990 determined the following bound. Theorem 4.7 ([305]) If 𝐺 is an isolate-free graph of order 𝑛, then  
Δ(𝐺) 1 𝛾(𝐺) ≤ 𝑛 + 1 − 𝛿(𝐺) − 1 . 2 𝛿(𝐺) An immediate corollary, due to Payan [632] in 1975, now follows. Corollary 4.8 ([632]) If 𝐺 is an isolate-free graph of order 𝑛, then 𝛾(𝐺) ≤
1 𝑛 + 2 − 𝛿(𝐺) . 2 In 1995 Slater [680] gave the following lower bound on the domination number of a graph that involves its nonincreasing degree sequence. The value of the lower bound is known as the Slater number sl(𝐺). Theorem 4.9 ([680]) If 𝐺 is a graph of order 𝑛 with degree sequence (𝑑1 , 𝑑2 , . . . , 𝑑 𝑛 ), where 𝑑1 ≥ 𝑑2 ≥ · · · ≥ 𝑑 𝑛 , then 𝛾(𝐺) ≥ sl(𝐺) = min 𝑘 : 𝑘 + (𝑑1 + 𝑑2 + · · · + 𝑑𝑘 ) ≥ 𝑛 . Proof Let 𝐷 be 𝛾-set of 𝐺, where |𝐷| = 𝑘. Each vertex 𝑣 ∈ 𝐷 is adjacent to at most deg(𝑣) vertices in 𝑉 \ 𝐷, while each vertex in 𝑉 \ 𝐷 has at least one neighbor in 𝐷. Hence, double counting the edges between 𝐷 and 𝑉 \ 𝐷,

Chapter 4. General Bounds

76 𝑛 − 𝑘 = |𝑉 \ 𝐷 | ≤

∑︁

deg(𝑣) ≤

𝑣 ∈𝐷

𝑘 ∑︁

𝑑𝑖 ,

𝑖=1

and so 𝑘 + (𝑑1 + 𝑑2 + · · · + 𝑑 𝑘 ) ≥ 𝑛. Hence, sl(𝐺) ≤ 𝑘 = |𝐷 | = 𝛾(𝐺).

4.2.2

Total Domination Number and Maximum Degree

An observation similar to Theorem 4.4 can be made for the total domination number. Recall that for sets of vertices 𝑋 and 𝑌 in a graph 𝐺 the set 𝑋 dominates 𝑌 if every vertex of 𝑌 has a neighbor in 𝑋 or belongs to 𝑋, and the set 𝑋 totally dominates 𝑌 if every vertex of 𝑌 has a neighbor in 𝑋. If 𝑋 dominates 𝑌 , 𝑋 = {𝑥}, and 𝑌 = {𝑦}, then we simply write that vertex 𝑥 totally dominates vertex 𝑦. In particular, if 𝑋 dominates the set 𝑉 (𝐺), then 𝑋 is a dominating set of 𝐺, and if 𝑋 totally dominates the set 𝑉 (𝐺), then 𝑋 is a TD-set of 𝐺. In a 𝛾t -set 𝑆 of a graph 𝐺, every vertex 𝑣 ∈ 𝑆 totally dominates at most Δ(𝐺) other vertices, namely all vertices in its open neighborhood N(𝑣). Thus, at most Δ(𝐺) vertices are totally dominated by each vertex of 𝑆, implying that at most Δ(𝐺)|𝑆| distinct vertices are totally dominated by the set 𝑆. Since every vertex is totally dominated by the set 𝑆, we therefore have that 𝑛 ≤ Δ(𝐺)|𝑆| = Δ(𝐺)𝛾t (𝐺). Further, if equality holds in this inequality chain, then the following conditions hold: (i) the set of open neighborhoods N(𝑣) of vertices 𝑣 ∈ 𝑆 is a partition of 𝑉 (𝐺), (ii) the subgraph 𝐺 [𝑆] is the disjoint union of copies of 𝐾2 , and (iii) each vertex of 𝑆 has degree Δ(𝐺). This yields the following elementary lower bound on the total domination number of a graph in terms of its order and maximum degree. Theorem 4.10 ([741]) If 𝐺 is a graph of order 𝑛, then 𝛾t (𝐺) ≥

𝑛 , Δ(𝐺)

with equality if and only if every 𝛾t -set in 𝐺 is a perfect open packing and every vertex in such a 𝛾t -set has degree Δ(𝐺). As an application of Theorem 4.10, we can readily determine the total domination number of a path 𝑃𝑛 and a cycle 𝐶𝑛 .       Proposition 4.11 For 𝑛 ≥ 2, 𝛾t (𝑃𝑛 ) = 𝑛2 + 𝑛4 − 𝑛4 , and for 𝑛 ≥ 3, 𝛾t (𝑃𝑛 ) = 𝛾t (𝐶𝑛 ). Proof Let 𝐺 = 𝐶𝑛 , where 𝑛 ≥ 3. The equality is immediate if 𝑛 = 3. Hence, we may assume that 𝐺 is the cycle 𝑣 1 𝑣 2 . . . 𝑣 𝑛 𝑣 1 , where 𝑛 ≥ 4. The elementary lower bound on the total domination number given in Theorem 4.10 yields 𝛾t (𝐺) ≥ 12 𝑛. Let 𝑆1 =

𝑛 ⌊Ø 4 ⌋−1

{𝑣 4𝑖+2 , 𝑣 4𝑖+3 }.

𝑖=0

We consider four cases based on 𝑛.

Section 4.2. Domination and Maximum Degree

77

Case 1. 𝑛 ≡ 0 (mod 4). Thus, 𝑛 = 4𝑘 for some 𝑘 ≥ 1. In this case, the set 𝑆1 is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ |𝑆1 | = 2𝑘 = 21 𝑛. Consequently, 𝛾t (𝐺) = 12 𝑛. Case 2. 𝑛 ≡ 2 (mod 4). Thus, 𝑛 = 4𝑘 + 2 for some 𝑘 ≥ 1. Suppose that 𝛾t (𝐺) = 12 𝑛 = 2𝑘 + 1. Let 𝐷 be a 𝛾t -set of 𝐺, and so |𝐷 | = 2𝑘 + 1 is odd. Thus, at least one component in 𝐺 [𝐷] has odd order at least 3, implying that at least one vertex in 𝐷 has no neighbor in 𝑉 \ 𝐷. Hence, counting edges between 𝐷 and 𝑉 \ 𝐷 yields at most |𝐷| − 1 edges. Since every vertex in 𝑉 \ 𝐷 has a neighbor in 𝐷, this yields an edge count between 𝐷 and 𝑉 \ 𝐷 of at least |𝑉 \ 𝐷| = 𝑛 − |𝐷|. Consequently, 𝑛 − |𝐷| ≤ |𝐷| − 1, and so |𝐷| ≥ 12 (𝑛 + 1), a contradiction. Therefore, 𝛾t (𝐺) > 12 𝑛 = 2𝑘 + 1, that is, 𝛾t (𝐺) ≥ 2𝑘 + 2. The set 𝑆2 = 𝑆1 ∪ {𝑣 4𝑘+1 , 𝑣 4𝑘+2 } is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ |𝑆2 | = 2𝑘 + 2. Consequently, 𝛾t (𝐺) ≤ 2𝑘 + 2 = 12 𝑛 + 1. Case 3. 𝑛 ≡ 1 (mod 4). Thus, 𝑛 = 4𝑘 + 1 for some 𝑘 ≥ 1. As observed earlier, 𝛾t (𝐺) ≥ 12 𝑛. Since 𝑛 is odd, we therefore have 𝛾t (𝐺) ≥ 12 (𝑛 + 1) = 2𝑘 + 1. The set 𝑆3 = 𝑆1 ∪ {𝑣 4𝑘 } is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ |𝑆3 | = 2𝑘 + 1. Consequently, 𝛾t (𝐺) = 2𝑘 + 1 = 12 (𝑛 + 1). Case 4. 𝑛 ≡ 3 (mod 4). Thus, 𝑛 = 4𝑘 + 3 for some 𝑘 ≥ 1. Since 𝑛 is odd, we have 𝛾t (𝐺) ≥ 12 (𝑛 + 1) = 2𝑘 + 2. The set 𝑆4 = 𝑆1 ∪ {𝑣 4𝑘+1 , 𝑣 4𝑘+2 } is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ |𝑆4 | = 2𝑘 + 2. Consequently, 𝛾t (𝐺) = 2𝑘 + 2 = 12 (𝑛 + 1).       In all cases, 𝛾t (𝐶𝑛 ) = 𝑛2 + 𝑛4 − 𝑛4 .  𝑛 An 𝑛identical   𝑛  construction of the earlier TD-set in a cycle 𝐶𝑛 yields 𝛾t (𝑃𝑛 ) ≤ + 2 4 − 4 . The desired result for a path now follows by applying Observation 4.2 to the result of a cycle. Suppose that a connected graph 𝐺 = (𝑉, 𝐸) of order 𝑛 ≥ 2 contains a dominating vertex 𝑣, that is, suppose 𝑉 = N[𝑣] and deg(𝑣) = Δ(𝐺) = 𝑛 − 1. In this case, the set consisting of 𝑣 and an arbitrary neighbor of 𝑣 form a TD-set of 𝐺, implying that 𝛾t (𝐺) = 2 = 𝑛 − Δ(𝐺) + 1. We state this formally as follows. Proposition 4.12 If 𝐺 is a connected graph of order 𝑛 ≥ 2 that contains a dominating vertex, then 𝛾t (𝐺) = 2 = 𝑛 − Δ(𝐺) + 1. In 1980 Cockayne et al. [182] established the following upper bound in terms of the order and maximum degree. Theorem 4.13 ([182]) If 𝐺 is a connected graph of order 𝑛 that does not contain a dominating vertex, then 𝛾t (𝐺) ≤ 𝑛 − Δ(𝐺). Proof Suppose that a connected graph 𝐺 of order 𝑛 does not contain a dominating vertex. Necessarily, 𝑛 ≥ 4. Let 𝑣 be a vertex of maximum degree in 𝐺, and so deg𝐺 (𝑣) = Δ(𝐺). Let 𝑇 be a spanning tree of 𝐺 that is distance-preserving from 𝑣. In particular, N𝐺 (𝑣) = N𝑇 (𝑣) and Δ(𝐺) = Δ(𝑇). Let 𝑇 have ℓ(𝑇) leaves. We note that ℓ(𝑇) ≥ Δ(𝑇). Let 𝑆 be the set of vertices that are not leaves of 𝑇. Since the graph 𝐺 does not contain a dominating vertex, removing all leaves from 𝑇 produces a connected graph of order at least 2. Thus, 𝐺 [𝑆] is a connected graph of order at least 2, implying that 𝑆 is a TD-set of 𝐺. Therefore, 𝛾t (𝐺) ≤ |𝑆| = 𝑛 − ℓ(𝑇) ≤ 𝑛 − Δ(𝑇) = 𝑛 − Δ(𝐺). As an application of Proposition 4.12, we have the following upper bound on the total domination number of a disconnected graph.

Chapter 4. General Bounds

78

Theorem 4.14 ([182]) If 𝐺 is a disconnected isolate-free graph of order 𝑛, then 𝛾t (𝐺) ≤ 𝑛 − Δ(𝐺) + 1, with equality if and only if every component of 𝐺 is a copy of 𝐾2 , except possibly for one component which contains a dominating vertex. Proof Let 𝐺 be a disconnected graph with components 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑘 where 𝑘 ≥ 2. We note that Δ(𝐺) = max Δ(𝐺 𝑖 ). 𝑖∈[𝑘]

Since 𝐺 is isolate-free, we note that Δ(𝐺 𝑖 ) ≥ 1 for all 𝑖 ∈ [𝑘]. Renaming components if necessary, we may assume that Δ(𝐺) = Δ(𝐺 1 ). With this assumption, 𝑘 ∑︁

Δ(𝐺 𝑖 ) = Δ(𝐺) +

𝑘 ∑︁

Δ(𝐺 𝑖 ) ≥ Δ(𝐺) + 𝑘 − 1.

(4.1)

𝑖=2

𝑖=1

By linearity and by Inequality (4.1), Proposition 4.12 and Theorem 4.13 yield the desired upper bound on the total domination number of 𝐺 as follows: 𝛾t (𝐺) =

𝑘 ∑︁

𝛾t (𝐺 𝑖 )

𝑖=1

≤

𝑘 ∑︁

𝑛(𝐺 𝑖 ) − Δ(𝐺 𝑖 ) + 1




𝑖=1

=

∑︁ 𝑘

 ∑︁  𝑘 Δ(𝐺 𝑖 ) + 𝑘 𝑛(𝐺 𝑖 ) −

𝑖=1

𝑖=1


≤ 𝑛 − Δ(𝐺) + 𝑘 − 1 + 𝑘 = 𝑛 − Δ(𝐺) + 1. Moreover, suppose that 𝛾t (𝐺) = 𝑛−Δ(𝐺) +1. Thus, we must have equality throughout the above inequality chain. In particular, 𝛾t (𝐺 1 ) = 𝑛(𝐺 1 ) − Δ(𝐺 1 ) + 1, implying by Proposition 4.12 and Theorem 4.13 that the component 𝐺 1 contains a dominating vertex. Furthermore, we must have equality in Inequality (4.1), implying that Δ(𝐺 𝑖 ) = 1 for all 𝑖 ∈ [𝑘] \ {1}, and so all such components 𝐺 𝑖 are 𝐾2 -components. For an example of graphs attaining the upper bound of Theorem 4.13, consider the subdivided star 𝐺 = 𝑆(𝐾1,𝑘 ) obtained from a star 𝐾1,𝑘 by subdividing every edge exactly once. The resulting graph 𝐺 has order 𝑛 = 2𝑘 + 1, Δ(𝐺) = 𝑘, and 𝛾t (𝐺) = 𝑘 + 1 = 𝑛 − Δ(𝐺). In 2001 Haynes and Markus [435] determined a property of the graphs attaining the upper bound of 𝑛 − Δ(𝐺) as follows. For a graph 𝐺 of order  𝑛 and 𝑘 ∈ [𝑛], the generalized maximum degree, denoted Δ 𝑘 (𝐺), of 𝐺 is the max |N(𝑆)| : 𝑆 ⊆ 𝑉 and |𝑆| = 𝑘 . Note that Δ1 (𝐺) = Δ(𝐺). Theorem 4.15 ([435]) If 𝐺 is a connected graph of order 𝑛 ≥ 3 with  Δ(𝐺) ≤ 𝑛 − 2, then 𝛾t (𝐺) = 𝑛−Δ(𝐺) if and only if Δ 𝑘 (𝐺) = Δ(𝐺) + 𝑘 for all 𝑘 ∈ 2, 3, . . . , 𝛾t (𝐺) .

Section 4.3. Domination and Order

79

Henning and Yeo noted in their book [490] that one direction of Theorem 4.15 can be strengthened as follows. of order 𝑛 ≥ 3 with Δ(𝐺) ≤ Theorem 4.16 ([490]) Let 𝐺 be a connected graph
𝑛 − 2. If Δ𝑛−Δ(𝐺) −1 (𝐺) = Δ(𝐺) + 𝑛 − Δ(𝐺) − 1 = 𝑛 − 1, then 𝛾t (𝐺) = 𝑛 − Δ(𝐺). We close this section with the following trivial lower bound on the total domination number that involves the nonincreasing degree sequence of a graph. This result has the same flavor as the Slater lower bound for domination given in Theorem 4.9. Theorem 4.17 If 𝐺 is an isolate-free graph of order 𝑛 with degree sequence (𝑑1 , 𝑑2 , . . . , 𝑑 𝑛 ) where 𝑑1 ≥ 𝑑2 ≥ · · · ≥ 𝑑 𝑛 , then 𝛾t (𝐺) ≥ min{𝑘 : 𝑑1 + 𝑑2 + · · · + 𝑑 𝑘 ≥ 𝑛}.

4.2.3 Independent Domination Number and Maximum Degree We establish next a trivial upper bound on the independent domination number of a graph in terms of its order and maximum degree. Let 𝑣 be a vertex of maximum degree Δ(𝐺) in a graph 𝐺, and let 𝐼 𝑣 be a maximal independent set that contains vertex 𝑣. Since 𝐼 𝑣 is an ID-set of 𝐺 that contains no neighbor of 𝑣, we have 𝑖(𝐺) ≤ |𝐼 𝑣 | ≤ |𝑉 \ N(𝑣)| = 𝑛 − deg(𝑣) = 𝑛 − Δ(𝐺). This yields the following upper bound on 𝑖(𝐺), due to Haviland [398] in 1991. Theorem 4.18 ([398]) If 𝐺 is a graph of order 𝑛, then 𝑖(𝐺) ≤ 𝑛 − Δ(𝐺).

4.3

Domination and Order

An obvious upper bound on the (independent/total) domination number of any graph is its order. And since every vertex must have a neighbor in a TD-set, an immediate lower bound on the total domination number is 2. We state this formally as follows. Observation 4.19 If 𝐺 is a graph of order 𝑛, then the following hold: (a) 1 ≤ 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝑛. (b) If 𝐺 is isolate-free, then 2 ≤ 𝛾t (𝐺) ≤ 𝑛. We note that the bounds of Observation 4.19 are tight. Clearly, for 𝑛 ≥ 2, if 𝐺 is a complete graph 𝐾𝑛 or a star 𝐾1,𝑛−1 , then 𝛾(𝐺) = 𝑖(𝐺) = 1 and 𝛾t (𝐺) = 2. On the other hand, for the empty graph 𝐺 = 𝐾 𝑛 , we have 𝛾(𝐺) = 𝑖(𝐺) = 𝑛. Similarly, for the graph 𝑘𝐾2 of order 𝑛 = 2𝑘 consisting of 𝑘 ≥ 1 vertex-disjoint copies of 𝐾2 , we have 𝛾t (𝑘𝐾2 ) = 𝑛. Observation 4.20 If 𝐺 is a graph of order 𝑛, then the following hold: (a) 𝛾(𝐺) = 1 (respectively, 𝑖(𝐺) = 1) if and only if 𝐺 has a dominating vertex. (b) 𝛾(𝐺) = 𝑛 (respectively, 𝑖(𝐺) = 𝑛) if and only if 𝐺 is the empty graph 𝐾 𝑛 .

Chapter 4. General Bounds

80

4.3.1 Domination Number and Order Recall that in Section 2.6, we presented Ore’s Lemmas [622] from 1962. In particular, Lemma 2.73 stated that the complement 𝑉 \ 𝐷 of any minimal dominating set 𝐷 in an isolate-free graph 𝐺 = (𝑉, 𝐸) is a dominating set. As animmediate consequence of this lemma, we have if 𝐺 has order 𝑛, then 𝛾(𝐺) ≤ min |𝐷|, 𝑛 − |𝐷| ≤ 21 𝑛. We state this formally as follows. Theorem 4.21 (Ore’s Theorem [622]) If 𝐺 is an isolate-free graph of order 𝑛, then 𝛾(𝐺) ≤ 12 𝑛. In 1979 Bollobás and Cockayne [84] established the following property of minimum dominating sets in graphs. Recall that for a set 𝑆 and a vertex 𝑣 ∈ 𝑆, the 𝑆-external private neighborhood of 𝑣 is abbreviated epn[𝑣, 𝑆] or epn(𝑣, 𝑆). Thus, epn[𝑣, 𝑆] = epn(𝑣, 𝑆). Lemma 4.22 ([84]) Every isolate-free graph 𝐺 contains a 𝛾-set 𝐷 such that epn[𝑣, 𝐷] ≠ ∅ for every vertex 𝑣 ∈ 𝐷. Proof Among all 𝛾-sets of 𝐺, let 𝐷 be chosen so that the number of edges in 𝐺 [𝐷] is a maximum. Let 𝑣 be an arbitrary vertex of 𝐷 and suppose, to the contrary, that epn[𝑣, 𝐷] = ∅. By Ore’s Lemma 2.72, ipn[𝑣, 𝐷] ≠ ∅, and so 𝑣 is an isolate in 𝐺 [𝐷] and therefore all neighbors of 𝑣 belong outside the set 𝐷. Since epn[𝑣, 𝐷] = ∅, every neighbor of 𝑣 is dominated by some vertex in 𝐷 \ {𝑣}. Replacing the vertex 𝑣 in 𝐷 by an arbitrary neighbor of 𝑣 outside 𝐷 produces a new 𝛾-set of 𝐺 that induces a subgraph containing more edges than does the subgraph induced by 𝐷, contradicting our choice of 𝐷. Hence, epn[𝑣, 𝐷] ≠ ∅. We note that if 𝐷 is a 𝛾-set of an isolate-free graph 𝐺 of order 𝑛 such that epn[𝑣, 𝐷] ≠ ∅ for every vertex 𝑣 ∈ 𝐷, then ∑︁ 𝑛 − |𝐷| = |𝑉 \ 𝐷| ≥ |epn[𝑣, 𝐷] | ≥ |𝐷 |, (4.2) 𝑣 ∈𝐷

and so |𝐷 | ≤ 12 𝑛. Hence, the bound of Theorem 4.21 also follows as an immediate consequence of Lemma 4.22. A vertex of degree 1 is called a leaf and the neighbor of a leaf is called a support vertex. We note that the set 𝑉 (𝐺) is a dominating set of the corona 𝐺 ◦ 𝐾1 , and so 𝛾(𝐺 ◦ 𝐾1 ) ≤ |𝑉 (𝐺)|. Moreover, every vertex of 𝐺 is a support vertex of 𝐺 ◦ 𝐾1 and either it or its leaf neighbor is in every 𝛾-set of 𝐺 ◦ 𝐾1 , implying that 𝛾(𝐺 ◦ 𝐾1 ) ≥ |𝑉 (𝐺)|. Consequently, 𝛾(𝐺 ◦ 𝐾1 ) = |𝑉 (𝐺)|. For example, the corona 𝐶5 ◦𝐾1 of the cycle 𝐶5 shown in Figure 4.1 has 𝛾(𝐶5 ◦𝐾1 ) = 5 and the five highlighted vertices form a 𝛾-set of 𝐶5 ◦ 𝐾1 . This yields the following observation. Observation 4.23 If 𝐺 is a corona 𝐻 ◦ 𝐾1 of order 𝑛, then 𝛾(𝐺) = 12 𝑛, and if 𝐻 is isolate-free, then 𝛾t (𝐺) = 12 𝑛. In 1982 Payan and Xuong [633] characterized the graphs achieving equality in Theorem 4.21, and showed that with the exception of a 4-cycle, coronas are the only

Section 4.3. Domination and Order

81

Figure 4.1 The corona 𝐶5 ◦ 𝐾1

graphs achieving equality in Ore’s Theorem 4.21. We remark that this characterization due to Payan and Xuong came 20 years after Ore’s Theorem. Theorem 4.24 ([633]) If 𝐺 is an isolate-free graph of even order 𝑛, then 𝛾(𝐺) = 12 𝑛 if and only if every component of 𝐺 is a 4-cycle or a corona 𝐻 ◦ 𝐾1 for some graph 𝐻. Proof Let 𝐺 be an isolate-free graph of even order 𝑛. Since the domination number of a graph is the sum of the domination numbers of its components and by Theorem 4.21 each component has domination number at most half its order, it suffices to prove the result when 𝐺 is connected. If 𝐺 = 𝐶4 , then 𝛾(𝐺) = 2 = 12 𝑛, and if 𝐺 = 𝐻 ◦ 𝐾1 for some graph 𝐻, then by Observation 4.23, we have 𝛾(𝐺) = 12 𝑛. Hence, the sufficiency is immediate. To prove the necessity, assume that 𝐺 is a connected graph of order 𝑛 satisfying 𝛾(𝐺) = 12 𝑛. Let 𝑘 = 12 𝑛 and let 𝐷 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } be a 𝛾-set of 𝐺 satisfying the statement of Lemma 4.22; that is, epn[𝑣 𝑖 , 𝐷] ≠ ∅ for every 𝑖 ∈ [𝑘]. Let 𝐷 = 𝑉 \ 𝐷. Since |𝐷 | = 12 𝑛, we must have equality throughout Inequality (4.2), implying that |epn[𝑣 𝑖 , 𝐷] | = 1 for every 𝑖 ∈ [𝑘] and 𝐷=

𝑘 Ø

epn[𝑣 𝑖 , 𝐷].

𝑖=1

Let epn[𝑣 𝑖 , 𝐷] = {𝑢 𝑖 } for 𝑖 ∈ [𝑘]. Thus, 𝐷 = {𝑢 1 , 𝑢 2 , . . . , 𝑢 𝑘 } and N𝐺 (𝑢 𝑖 )∩𝐷 = {𝑣 𝑖 } for 𝑖 ∈ [𝑘], implying that the set of edges between 𝐷 and 𝐷 is the set [𝐷, 𝐷] =

𝑘 Ø

{𝑢 𝑖 𝑣 𝑖 }.

𝑖=1

If 𝑛 = 2, then 𝐺 = 𝐾2 = 𝐾1 ◦ 𝐾1 . If 𝑛 = 4, then either 𝐺 = 𝐶4 or 𝐺 = 𝑃4 = 𝐾2 ◦ 𝐾1 . Hence, we may assume that 𝑛 ≥ 6, for otherwise the desired result follows. Thus, 𝑘 = 12 𝑛 ≥ 3. We show first that exactly one of 𝑢 𝑖 and 𝑣 𝑖 has degree 1 in 𝐺 for all 𝑖 ∈ [𝑘]. Renaming vertices if necessary, suppose, to the contrary, that both 𝑢 1 and 𝑣 1 have degree at least 2 in 𝐺. In this case, 𝑣 1 𝑣 𝑖 ∈ 𝐸 (𝐺) and 𝑢 1 𝑢 𝑗 ∈ 𝐸 (𝐺) for some 𝑖 ∈ [𝑘] \ {1} and some 𝑗 ∈ [𝑘] \ {1}. If 𝑖 ≠ 𝑗, then 𝑣 𝑗 𝑢 𝑗 𝑢 1 𝑣 1 𝑣 𝑖 𝑢 𝑖 is a path 𝑃6 in 𝐺. If 𝑖 = 𝑗, then 𝑢 1 𝑣 1 𝑣 𝑖 𝑢 𝑖 𝑢 1 is an (induced) 4-cycle 𝐶 in 𝐺. Since 𝐺 is a connected graph of order 𝑛 ≥ 6, there is an edge in 𝐺 joining a vertex in 𝐶 to a vertex outside 𝐶. Renaming vertices if necessary, we may assume that 𝑣 1 𝑣 𝑘 is such an edge, and so

82

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𝑢 𝑘 𝑣 𝑘 𝑣 1 𝑢 1 𝑢 𝑖 𝑣 𝑖 is a path
𝑃6 in 𝐺. In both cases, 𝐺 contains a spanning subgraph 𝐻 isomorphic to 𝑃6 ∪ 𝑛−6 2 𝐾2 . Since adding edges to a graph does not increase its domination number, 𝛾(𝐺) ≤ 𝛾(𝐻) = 𝛾(𝑃6 ) + 𝑛−6 2 𝛾(𝐾2 ) = 2 + (𝑛 − 6)/2 < 𝑛/2, a contradiction. Hence, exactly one of 𝑢 𝑖 and 𝑣 𝑖 has degree 1 in 𝐺 for all 𝑖 ∈ [𝑘]. Let 𝑋 be the set of vertices of degree at least 2 in 𝐺, and let 𝑌 = 𝑉 \ 𝑋. We note that |𝑋 | = |𝑌 | = 12 𝑛. Further, 𝑌 is an independent set and each vertex in 𝑌 has degree 1 in 𝐺. Since 𝐺 is a connected graph, 𝐺 = 𝐻 ◦ 𝐾1 , where 𝐻 = 𝐺 [𝑋] is the connected graph induced by the set 𝑋. In 1998 Randerath and Volkmann [648], and independently in 2000 Baogen et al. [57], characterized the isolate-free graphs 𝐺 of odd order 𝑛 satisfying 𝛾(𝐺) = 1 2 (𝑛 − 1). A proof of this result can be found in [417]. In Chapter 6, we present improved upper bounds on the domination number of a connected graph 𝐺 in terms of its order 𝑛 when the minimum degree is at least 2. For example, in Chapter 6 we present the 1989 result of McCuaig and Shepherd [586] that if 𝛿(𝐺) ≥ 2 and 𝑛 ≥ 8, then 𝛾(𝐺) ≤ 25 𝑛, and the 1996 result of Reed [655] that if 𝛿(𝐺) ≥ 3, then 𝛾(𝐺) ≤ 38 𝑛.

4.3.2

Total Domination Number and Order

A fundamental property of minimal TD-sets was established by Cockayne et al. [182] in 1980. Lemma 4.25 ([182]) A TD-set 𝑆 in a graph 𝐺 is a minimal TD-set if and only if every vertex 𝑣 ∈ 𝑆 has an open 𝑆-external private neighbor or an open 𝑆-internal private neighbor, that is, if and only if |epn(𝑣, 𝑆)| ≥ 1 or |ipn(𝑣, 𝑆)| ≥ 1. Proof Let 𝑆 be a minimal TD-set in 𝐺 and let 𝑣 ∈ 𝑆. If |epn(𝑣, 𝑆)| = 0 and |ipn(𝑣, 𝑆)| = 0, then every vertex in 𝐺 has a neighbor in 𝑆 \ {𝑣}, implying that 𝑆 \ {𝑣} is a TD-set of 𝐺, contradicting the minimality of 𝑆. Therefore, |epn(𝑣, 𝑆)| ≥ 1 or |ipn(𝑣, 𝑆)| ≥ 1 for every vertex 𝑣 ∈ 𝑆. Conversely, if |epn(𝑣, 𝑆)| ≥ 1 or |ipn(𝑣, 𝑆)| ≥ 1 for each 𝑣 ∈ 𝑆, then 𝑆 \ {𝑣} is not a TD-set, implying that 𝑆 is a minimal TD-set in 𝐺. The following stronger property of a minimum TD-set in a graph was established in 2000 by Henning [453]. Lemma 4.26 ([453]) If 𝐺 ≠ 𝐾𝑛 is a connected graph of order 𝑛 ≥ 3, then 𝐺 has a 𝛾t -set 𝑆 such that every vertex 𝑣 ∈ 𝑆 has an open 𝑆-external private neighbor or has an open 𝑆-internal private neighbor which in turn has an open 𝑆-external private neighbor, that is, for every vertex 𝑣 ∈ 𝑆, we have |epn(𝑣, 𝑆)| ≥ 1 or there exists a vertex 𝑣 ′ ∈ ipn(𝑣, 𝑆) with |epn(𝑣 ′ , 𝑆)| ≥ 1. Proof Let 𝐺 ≠ 𝐾𝑛 be a connected graph of order 𝑛 ≥ 3. Among all 𝛾t -sets of 𝐺, let 𝑆 be chosen so that (i) the number of edges in 𝐺 [𝑆] is a maximum and (ii) subject ′ to (i), the number of vertices 𝑣 ∈ 𝑆 having |epn(𝑣, 𝑆)|  ≥ 1 or |epn(𝑣 , 𝑆)| ≥ 1 for ′ some neighbor 𝑣 ∈ 𝑆 of 𝑣 is a maximum. Let 𝐴 = 𝑣 ∈ 𝑆 : |epn(𝑣, 𝑆)| ≥ 1 , and

Section 4.3. Domination and Order

83

let 𝐵 be a minimum set of vertices of 𝑆 \ 𝐴 such that 𝐺 [ 𝐴 ∪ 𝐵] has no isolated vertices. Necessarily, |𝐵| ≤ | 𝐴|. Further, let 𝐶 = 𝑆 \ ( 𝐴 ∪ 𝐵). We show that 𝐶 = ∅. Suppose, to the contrary, that 𝐶 ≠ ∅. Let 𝐹 = 𝐺 [𝐶]. Since epn(𝑣, 𝑆) = ∅ for every vertex 𝑣 ∈ 𝐶, by Lemma 4.25, each such vertex 𝑣 satisfies |ipn(𝑣, 𝑆)| ≥ 1. Further, since 𝐺 [ 𝐴 ∪ 𝐵] contains no isolated vertex, we deduce that 𝐹 =𝑘𝐾2 for some 𝑘 ≥ 1 and that each vertex of 𝐶 has degree 1 in 𝐹. Let 𝐸 (𝐹) = 𝑢 𝑖 𝑣 𝑖 : 𝑖 ∈ [𝑘] . Further, let 𝑁𝑖 = N𝐺 ({𝑢 𝑖 , 𝑣 𝑖 }) \ {𝑢 𝑖 , 𝑣 𝑖 } for all 𝑖 ∈ [𝑘], and let 𝑘 Ø 𝑁= 𝑁𝑖 . 𝑖=1

Since 𝐺 is connected and 𝑛 ≥ 3, we note that the set 𝑁𝑖 ≠ ∅ for 𝑖 ∈ [𝑘]. Let 𝑤 𝑖 ∈ 𝑁𝑖 for 𝑖 ∈ [𝑘]. Since each vertex of 𝐶 has degree 1 in 𝐺 [𝑆], we note that 𝑁𝑖 ⊆ 𝑉 \ 𝑆. Renaming 𝑢 𝑖 and 𝑣 𝑖 if necessary, we may assume that 𝑣
𝑖 𝑤 𝑖 ∈ 𝐸 (𝐺). If 𝑤 𝑖 is adjacent to a vertex of 𝑆 different from 𝑢 𝑖 and 𝑣 𝑖 , then 𝑆 \ {𝑢 𝑖 } ∪ {𝑤 𝑖 } is a 𝛾t -set of 𝐺 whose induced subgraph contains more edges than the subgraph induced by 𝑆, contradicting our choice of 𝑆. Hence, no vertex in 𝑁𝑖 has a neighbor in 𝑆 \ {𝑢 𝑖 , 𝑣 𝑖 }. Thus, the sets 𝑁1 , 𝑁2 , . . . , 𝑁 𝑘 are pairwise disjoint subsets. Since epn(𝑢 𝑖 , 𝑆) = epn(𝑣 𝑖 , 𝑆) = ∅, each vertex in 𝑁𝑖 is adjacent to both 𝑢 𝑖 and 𝑣 𝑖 for all 𝑖 ∈ [𝑘]. Suppose that 𝑆 = {𝑢 1 , 𝑣 1 }. In this case, since 𝐺 ≠ 𝐾𝑛 and 𝑉 = 𝑁1 ∪ {𝑢 1 , 𝑣 1 }, the set 𝑁1 contains two nonadjacent vertices. We may assume that 𝑤 1 ∈ 𝑁1 is not adjacent to some vertex of 𝑁1 . The set 𝑆 ′ = {𝑣 1 , 𝑤 1 } is a 𝛾t -set of 𝐺 such that |epn(𝑣 1 , 𝑆 ′ )| ≥ 1, contradicting our choice of the set 𝑆. Hence 𝑆 ≠ {𝑢 1 , 𝑣 1 }. Thus, since 𝐺 is connected, each set 𝑁𝑖 contains a vertex that is adjacent to a vertex of 𝑉 \ (𝑆 ∪ 𝑁𝑖 ) for all 𝑖 ∈ [𝑘]. We may assume 𝑤 𝑖 is such a vertex of 𝑁𝑖 . We show next that there is no edge joining a vertex of 𝑁𝑖 and a vertex of 𝑁 𝑗 , vertices if necessary, where 𝑖, 𝑗 ∈ [𝑘] and 𝑖 ≠ 𝑗. If this is not the case, then renaming
we may assume 𝑤 1 𝑤 2 ∈ 𝐸 (𝐺). Thus, 𝑆 \ {𝑢 1 , 𝑢 2 } ∪ {𝑤 1 , 𝑤 2 } is a 𝛾t -set of 𝐺 whose induced subgraph contains more edges than the subgraph induced by 𝑆, a contradiction. Hence, there is no edge joining a vertex of 𝑁𝑖 and a vertex of 𝑁 𝑗 where 𝑖, 𝑗 ∈ [𝑘] and 𝑖 ≠ 𝑗, implying that each vertex 𝑤 𝑖 is adjacent to a vertex, say 𝑦 𝑖 , of 𝑉 \ (𝑆 ∪ 𝑁) for all 𝑖 ∈ [𝑘]. Further, we note that such a vertex 𝑦 𝑖 has no neighbor in 𝐶 and therefore has at least one neighbor in 𝐴 ∪ 𝐵.
If 𝑤 𝑖 is adjacent to every other vertex of 𝑁𝑖 for some 𝑖 ∈ [𝑘], then 𝑆 \ {𝑢 𝑖 , 𝑣 𝑖 } ∪ {𝑤 𝑖 , 𝑦 𝑖 } is a 𝛾t -set of 𝐺 whose induced subgraph contains more edges than the subgraph induced by 𝑆, a contradiction. Hence, 𝑤 𝑖 is not adjacent to at least one vertex, say 𝑥𝑖 , of 𝑁𝑖 for all 𝑖 ∈ [𝑘].
We now consider the set 𝑆 ′ = 𝑆 \ {𝑢 1 } ∪ {𝑤 1 }. Necessarily, 𝑆 ′ is a 𝛾t -set of 𝐺. Since 𝑥 1 ∈ epn(𝑣 1 , 𝑆 ′ ), we note that |epn(𝑣 1 , 𝑆 ′ )| ≥ 1. If every  vertex 𝑎 ∈ 𝐴 satisfies ′ )| ≥ 1, then 𝐺 [𝑆 ′ ] has the same size as 𝐺 [𝑆] but 𝑣 ∈ 𝑆 ′ : |epn(𝑣, 𝑆 ′ )| ≥ |epn(𝑎, 𝑆  1 > 𝑣 ∈ 𝑆 : |epn(𝑣, 𝑆)| ≥ 1 , contradicting our choice of the set 𝑆. Hence, there exists a vertex 𝑎 ∈ 𝐴 such that epn(𝑎, 𝑆 ′ ) = ∅, implying that vertex 𝑤 1 dominates the set epn(𝑎, 𝑆). By Lemma 4.25, |ipn(𝑎, 𝑆 ′ )| ≥ 1. Let 𝑎 ′ ∈ epn(𝑎, 𝑆) and let 𝑎★ ∈ ipn(𝑎, 𝑆 ′ ). We note that vertex 𝑎 is the only vertex in 𝑆 ′ that is adjacent to 𝑎★, and so the vertex 𝑎★ has degree 1 in 𝐺 [𝑆 ′ ]. If 𝑎★ ∈ 𝐵 or if 𝑎★ ∈ 𝐴 and

Chapter 4. General Bounds

84


epn(𝑎★, 𝑆 ′ ) = ∅, then 𝑆 ′ \ {𝑎 ∗ } ∪ {𝑎 ′ } is a 𝛾t -set of 𝐺 whose induced subgraph contains more edges than the subgraph induced by 𝑆, a contradiction. Hence, 𝑎★ ∈ 𝐴. Thus, vertex 𝑎 is adjacent to a vertex 𝑎★ ∈ 𝑆 ′ such that |epn(𝑎★, 𝑆 ′ )| ≥ 1. This implies that 𝐺 [𝑆 ′ ] has the same size as 𝐺 [𝑆], but there are more vertices 𝑣 ∈ 𝑆 ′ such that |epn(𝑣, 𝑆 ′ )| ≥ 1 or |epn(𝑣 ′ , 𝑆 ′ )| ≥ 1 for some neighbor 𝑣 ′ ∈ 𝑆 ′ of 𝑣 than there are vertices in 𝑆, once again contradicting our choice of the set 𝑆. Therefore, 𝐶 = ∅. By our choice of the set 𝐵, every vertex in 𝐵 has a neighbor in 𝐴. Let 𝑤 ∈ 𝐵, and so epn(𝑤, 𝑆) = ∅. By Lemma 4.25, we note that |ipn(𝑤, 𝑆)| ≥ 1. Let 𝑣 ∈ ipn(𝑤, 𝑆). Since every vertex in 𝐵 has a neighbor in 𝐴, we note that 𝑣 ∉ 𝐵, and so 𝑣 ∈ 𝐴, implying that the vertex 𝑣 has degree 1 in 𝐺 [𝑆]. Since 𝑤 is an arbitrary vertex in 𝐵, this shows that every vertex in 𝐵 has a neighbor 𝑣 ∈ 𝑆 of degree 1 in 𝐺 [𝑆] such that |epn(𝑣, 𝑆)| ≥ 1. Cockayne et al. [182] proved that the total domination of a connected graph of order at least 3 is at most two-thirds its order. The proof follows from a simple application of Lemma 4.26. Theorem 4.27 ([182]) If 𝐺 is a connected graph of order 𝑛 ≥ 3, then 𝛾t (𝐺) ≤ 23 𝑛. Proof Let 𝐺 be a connected graph of order 𝑛 ≥ 3. If 𝐺 = 𝐾𝑛 , then 𝛾t (𝐺) = 2 ≤ 23 𝑛. Hence, we may assume that 𝐺 ≠ 𝐾𝑛 . Let 𝑆 be a 𝛾t -set of 𝐺 satisfying the statement of Lemma 4.26. Let 𝐴 = 𝑣 ∈ 𝑆 : |epn(𝑣, 𝑆)| ≥ 1 and let 𝐵 = 𝑆 \ 𝐴. Thus,  𝐵 = 𝑣 ∈ 𝑆 : epn(𝑣, 𝑆) = ∅ , and so, by Lemma 4.26, each vertex 𝑢 ∈ 𝐵 has a neighbor 𝑣 ∈ 𝐴 of degree 1 in 𝐺 [𝑆], and so |epn(𝑣, 𝑆)| ≥ 1 and the vertex 𝑢 is the unique neighbor of 𝑣 in 𝐺 [𝑆]. Thus, |𝐵| ≤ | 𝐴|, and so |𝑆| = | 𝐴| + |𝐵| ≤ 2| 𝐴|. Let 𝐶 be the set of all external 𝑆-private neighbors, and so 𝐶 ⊆ 𝑉 \ 𝑆. Thus, Ø ∑︁ 𝐶= epn(𝑣, 𝑆) and |𝐶 | = |epn(𝑣, 𝑆)| ≥ | 𝐴|. 𝑣∈ 𝐴

𝑣∈ 𝐴

Hence, 𝑛 − |𝑆| = |𝑉 \ 𝑆| ≥ |𝐶 | ≥ | 𝐴| ≥ 12 |𝑆|,

(4.3)

and so 𝛾t (𝐺) = |𝑆| ≤ 23 𝑛. The 2-corona 𝐹 ◦ 𝑃2 of a connected graph 𝐹 is the graph of order 3|𝑉 (𝐹)| obtained from 𝐹 by attaching a path of length 2 to each vertex of 𝐹 so that the resulting paths are vertex-disjoint. For example, the 2-corona 𝐶4 ◦ 𝑃2 of a 4-cycle is illustrated in Figure 4.2. We note that every TD-set of the 2-corona 𝐹 ◦ 𝑃2 of a graph 𝐹 contains all support vertices of 𝐹 ◦ 𝑃2 . Moreover in order to totally dominate the support vertices, every TD-set also contains a neighbor of each support vertex. Thus, if 𝐹 has order 𝑘, then 𝛾t (𝐹 ◦ 𝑃2 ) ≥ 2𝑘, noting that 𝐹 has 𝑘 support vertices. The set 𝑉 (𝐹), together with the set of all support vertices of 𝐹 ◦ 𝑃2 , is a TD-set of 𝐹 ◦ 𝑃2 , and so 𝛾t (𝐹 ◦ 𝑃2 ) ≤ 2𝑘. Consequently, 𝛾t (𝐹 ◦ 𝑃2 ) = 2𝑘. In Figure 4.2, 𝛾t (𝐶4 ◦ 𝑃2 ) = 8 and the eight highlighted vertices, for example, form a 𝛾t -set of 𝐶4 ◦ 𝑃2 . This yields the following observation. Observation 4.28 If 𝐺 is a 2-corona 𝐹 ◦ 𝑃2 of order 𝑛, then 𝛾t (𝐺) = 23 𝑛.

Section 4.3. Domination and Order

85

Figure 4.2 The 2-corona 𝐶4 ◦ 𝑃2

In 2000 Brigham et al. [117] characterized the connected graphs that achieve equality in the upper bound of Theorem 4.27, and showed that, with the exception of two small cycles, the extremal graphs are precisely the family of 2-coronas of connected graphs. The proof we present of their characterization follows readily from Lemma 4.26. Theorem 4.29 ([117]) If 𝐺 is a connected graph of order 𝑛 ≥ 3, then 𝛾t (𝐺) = 23 𝑛 if and only if 𝐺 is 𝐶3 , 𝐶6 , or 𝐹 ◦ 𝑃2 for some connected graph 𝐹. Proof Let 𝐺 be a connected graph of order 𝑛 ≥ 3. If 𝐺 = 𝐾𝑛 and 𝛾t (𝐺) = 23 𝑛, then 𝛾t (𝐺) = 2 = 23 𝑛, implying that 𝑛 = 3 and 𝐺 is the cycle 𝐶3 . Hence, we may assume that 𝐺 ≠ 𝐾𝑛 . Adopting the notation in the proof of Theorem 4.27, if 𝛾t (𝐺) = 23 𝑛, then there must be equality throughout Inequality (4.3). In particular, | 𝐴| = |𝐵| = |𝐶 | = 13 𝑛 and 𝑉 \ 𝑆 = 𝐶, implying that each vertex of 𝐴 has degree 2 in 𝐺 and has one neighbor in 𝐵 and its other neighbor in 𝐶. Thus, the graph 𝐺 contains a spanning subgraph 𝐻 = 𝑘 𝑃3 , where 𝑘 = 13 𝑛. Suppose that both sets 𝐵 and 𝐶 contain a vertex of degree 2 or more in 𝐺. In this case, by the connectivity of 𝐺, the graph 𝐺 contains a spanning subgraph 𝐻, where 𝐻 = 𝐶6 or 𝐻 = 𝑃9 ∪ (𝑘 − 3)𝑃3 . If 𝐻 = 𝑃9 ∪ (𝑘 − 3)𝑃3 , then 𝛾t (𝐻) = 5 + 2(𝑘 − 3) = 2𝑘 − 1 < 23 𝑛. Since adding edges to a graph does not increase the total domination number, this would imply that 𝛾t (𝐺) ≤ 𝛾t (𝐻) < 23 𝑛, a contradiction. Hence, 𝐻 = 𝐶6 and 𝐺 contains no spanning subgraph isomorphic to 𝑃9 ∪ (𝑘 − 3)𝑃3 . This implies that 𝐺 = 𝐻 = 𝐶6 . Hence, we may assume that one of the sets 𝐵 or 𝐶 contains only vertices of degree 1 in 𝐺. If every vertex of 𝐵 has degree 1 in 𝐺, then by the connectivity of 𝐺, the subgraph 𝐺 [𝐶] is connected and 𝐺 = 𝐹 ◦ 𝑃2 where 𝐹 = 𝐺 [𝐶], while if every vertex of 𝐶 has degree 1 in 𝐺, then the subgraph 𝐺 [𝐵] is connected and 𝐺 = 𝐹 ◦ 𝑃2 where 𝐹 = 𝐺 [𝐵]. In both cases, 𝐺 = 𝐹 ◦ 𝑃2 for some connected graph 𝐹. The total domination number of a graph is equal to the sum of the total domination numbers of its components. Hence, as a consequence of Theorem 4.27, we have the following trivial characterization of graphs having total domination number equal to their order. Corollary 4.30 If 𝐺 is an isolate-free graph of order 𝑛, then 𝛾t (𝐺) = 𝑛 if and only if 𝐺 = 𝑘𝐾2 for some 𝑘 ≥ 1.

Chapter 4. General Bounds

86

In Chapter 6, we present improved upper bounds on the total domination number of a connected graph 𝐺 in terms of its order 𝑛, when the minimum degree is at least 2. For example, we show in this chapter that if 𝛿(𝐺) ≥ 2 and 𝑛 ≥ 11, then 𝛾t (𝐺) ≤ 74 𝑛. We will also prove among other bounds the result that if 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 12 𝑛, and if 𝛿(𝐺) ≥ 4, then 𝛾t (𝐺) ≤ 37 𝑛.

4.3.3

Independent Domination Number and Order

In this section, we present two upper bounds on the independent domination number of a graph in terms of its order. Using the property of a minimum dominating set of a graph established in Lemma 4.22, Bollobás and Cockayne [84] in 1979 proved the following upper bound on the independent domination number. Theorem 4.31 ([84]) If 𝐺 is an isolate-free graph of order 𝑛, then 𝑖(𝐺) ≤ 𝑛 + 2 − 𝛾(𝐺) −

𝑛 . 𝛾(𝐺)

Proof By Lemma 4.22, there exists a 𝛾-set 𝐷 of any isolate-free graph 𝐺 such that epn(𝑣, 𝐷) ≠ ∅ for every vertex 𝑣 ∈ 𝐷. For each vertex 𝑣 ∈ 𝐷, choose an arbitrary vertex 𝑣 ′ ∈ epn(𝑣, 𝐷). Let 𝛾 = 𝛾(𝐺). By
the Pigeonhole Principle, there is a vertex 𝑣 ∈ 𝐷 that is adjacent to at least 𝑛 − |𝐷| / |𝐷| = (𝑛 − 𝛾)/𝛾 vertices  of 𝑉 \ 𝐷. Let 𝑁 𝑣 be the set of neighbors of 𝑣 that belong to 𝑉 \ 𝐷, and so |𝑁 𝑣 | ≥ (𝑛 − 𝛾)/𝛾 . Let 𝐼 be a maximal independent set in 𝐺 that contains the vertex 𝑣. We note that 𝐼 ∩ 𝑁 𝑣 = ∅. Further, the set 𝐼 contains at most one of 𝑢 and 𝑢 ′ for every vertex 𝑢 ∈ 𝐷 \ {𝑣}, and so there are at least |𝐷| − 1 vertices not in 𝑁 𝑣 that do not belong to the set 𝐼. Therefore,
|𝐼 | ≤ 𝑛 − |𝐷| − 1 − |𝑁 𝑣 |   𝑛−𝛾 ≤ 𝑛 − (𝛾 − 1) − 𝛾   𝑛 =𝑛+2−𝛾− 𝛾 𝑛 ≤ 𝑛+2−𝛾− . 𝛾 Since 𝐼 is a maximal ID-set of 𝐺, it follows that 𝑖(𝐺) ≤ |𝐼 |. √ Treating 𝑛 as fixed, the function 𝑓 (𝛾) = 𝑛 + 2 − 𝛾 − 𝛾𝑛 is maximized at 𝛾 = 𝑛. √
√ Thus since 𝑓 𝑛 = 𝑛 + 2 − 2 𝑛, as an immediate consequence of Theorem 4.31, we obtain the following bound, first observed in 1988 by Favaron [274] (and also proved in 1995 by Gimbel and Vestergaard [336]). Theorem 4.32 ([274]) If 𝐺 is an isolate-free graph of order 𝑛, then √ 𝑖(𝐺) ≤ 𝑛 + 2 − 2 𝑛.

Section 4.4. Basic Relationships Among Core Parameters

87

That the bound of Theorem 4.32 is tight may be seen as follows. For any integer 𝑟 ≥ 1, the generalized corona cor(𝐺, 𝑟) of a graph 𝐺 is the graph obtained from 𝐺 by adding 𝑟 pendant edges to each vertex of 𝐺, that is, for each vertex 𝑣 of 𝐺, we add 𝑟 new vertices and add an edge from each new vertex to the vertex 𝑣. In particular, if 𝑟 = 1, then cor(𝐺, 𝑟) is the corona 𝐺 ◦ 𝐾1 (also denoted cor(𝐺)). For example, if 𝐺 is the cycle 𝐶5 and 𝑟 = 3, then the generalized corona cor(𝐺, 𝑟) of 𝐺 is illustrated in Figure 4.3.

Figure 4.3 The generalized corona cor(𝐶5 , 3) For 𝑟 ≥ 2, if we take 𝐺 √= cor(𝐾𝑟 , 𝑟 − 1), then 𝐺 has order 𝑛 = 𝑟 2 and 𝑖(𝐺) = (𝑟 − 1) 2 + 1 = 𝑛 + 2 − 2 𝑛, showing that the upper bound of Theorem 4.32 is tight. This implies that, unlike the domination and total domination numbers, there is no constant 𝑘 < 1 such that for every isolate-free graph 𝐺 of order 𝑛, 𝑖(𝐺) ≤ 𝑘𝑛. Brigham et al. [117] investigated the graphs that attain (the floor of) the bound in Theorem 4.32. In particular, they showed that if 𝑛 is a square, then the generalized coronas cor(𝐾𝑟 , 𝑟 − 1) given above are the only extremal graphs. Thus, in this case all extremal graphs achieving the bound of Theorem 4.32 have minimum degree 1. In Chapter 6, we present improved upper bounds on the independent domination number of a connected graph 𝐺 in terms of its order 𝑛 when the minimum degree is at least 2. In particular, we will prove a much stronger result than Theorem 4.32, namely √ that if 𝐺 is a graph of order 𝑛 with minimum degree 𝛿, then 𝑖(𝐺) ≤ 𝑛 + 2𝛿 − 2 𝛿𝑛.

4.4

Basic Relationships Among Core Parameters

As mentioned in Chapter 2, Cockayne et al. [196] were the first to observe the important Domination Chain relating the core parameters 𝛾, 𝑖, 𝛼, and Γ. Theorem 4.33 ([196]) For every graph 𝐺, 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝛼(𝐺) ≤ Γ(𝐺). In 1979 Bollobás and Cockayne [84] were the first to observe the following relationship between the domination and total domination numbers of an isolate-free graph. Theorem 4.34 ([84]) If 𝐺 is an isolate-free graph, then 𝛾(𝐺) ≤ 𝛾t (𝐺) ≤ 2𝛾(𝐺). Theorems 4.33 and 4.34 establish the fundamental relationships among the core domination parameters. In Chapter 15, we revisit these results, present a proof of Theorem 4.34, and explore other relationships among these parameters.

88

Chapter 4. General Bounds

4.5 Domination and Distance Our next result gives a lower bound on the domination number in terms of diameter. Theorem 4.35 If 𝐺 is a connected graph, then 1
 3 diam(𝐺) + 1 ≤ 𝛾(𝐺). Proof Let 𝐺 be a connected graph with diam(𝐺) = 𝑑, and let 𝑆 be a 𝛾-set of 𝐺. Let 𝑃 : 𝑣 1 𝑣 2 . . . 𝑣 𝑑+1 be a path such that 𝑑 (𝑣 1 , 𝑣 𝑑+1 ) = diam(𝐺) = 𝑑. We note that any vertex on 𝑃 dominates at most three vertices on 𝑃. Suppose that there is a vertex 𝑢 ∈ 𝑆, such that 𝑢 is not on 𝑃 and 𝑢 dominates at least four vertices on 𝑃. Thus, N(𝑢) contains two vertices 𝑣 𝑖 and 𝑣 𝑗 on 𝑃 such that 𝑑 (𝑣 𝑖 , 𝑣 𝑗 ) ≥ 3 and 𝑖 < 𝑗. However, replacing the subpath on 𝑃 from 𝑣 𝑖 to 𝑣 𝑗 with the path 𝑣 𝑖 𝑢 𝑣 𝑗 creates a path 𝑃′ : 𝑣 1 𝑣 2 . . . 𝑣 𝑖 𝑢 𝑣 𝑗 𝑣 𝑗+1 . . . 𝑣 𝑑+1 from 𝑣 1 to 𝑣 𝑑+1 that is shorter than 𝑃, contradicting the fact that 𝑃 is a shortest path from 𝑣 1 to 𝑣 𝑑+1 . Hence, every  vertex  in 𝑆 dominates
 at most three vertices of 𝑃, implying that 𝛾(𝐺) = |𝑆| ≥ 13 |𝑃| = 13 diam(𝐺) + 1 . Theorem 4.35 is tight as can be seen by the following result.   
 Proposition 4.36 For 𝑛 ≥ 3, 𝛾(𝑃𝑛 ) = 𝑛3 = 13 diam(𝑃𝑛 ) + 1 . A graph 𝐺 for which diam(𝐺) = 2 is called a diameter-2 graph. For a diameter-2 graph 𝐺 and any vertex 𝑣 of 𝐺, the open neighborhood N(𝑣) of 𝑣 dominates 𝐺, while the closed neighborhood N[𝑣] of 𝑣 totally dominates 𝐺. Choosing such a vertex 𝑣 of minimum degree immediately yields the following observation. Observation 4.37 If 𝐺 is a diameter-2 graph, then 𝛾(𝐺) ≤ 𝛿(𝐺) and 𝛾t (𝐺) ≤ 𝛿(𝐺) + 1. We remark that if 𝐺 is a star 𝐾1,𝑘 where 𝑘 ≥ 2, then 𝐺 is a diameter-2 graph satisfying 𝛾(𝐺) = 𝛿(𝐺) = 1 and 𝛾t (𝐺) = 𝛿(𝐺) + 1 = 2. If 𝐺 is a diameter-2 graph with 𝛿(𝐺) = 2 that does not contain a dominating vertex, then 𝛾(𝐺) = 𝛿(𝐺) = 2. In Chapter 7, we study the (total) domination number of a diameter-2 graph in more depth. For example, we will show that given any 𝜀 > 0, if 𝐺 is
√︁a diameter-2 graph of sufficiently large order 𝑛, then 𝛾(𝐺) ≤ 𝛾t (𝐺) < √1 + 𝜀 𝑛 ln(𝑛). We 2 will also show that if we choose the probability 𝑝 carefully, then any random graph in G(𝑛, 𝑝) (which we will define formally in Section 7.3) is a diameter-2 graph with√︁its domination √︁and total domination numbers concentrated between roughly 1 √ 𝑛 ln(𝑛) and √1 𝑛 ln(𝑛). 2 2 2 Let diam(𝐺) = ∞ for a disconnected graph. Since any pair of vertices at distance 3 or more apart in 𝐺 form a dominating set of its complement 𝐺, we have the following observation made by Brigham et al. [118] in 1988. Observation 4.38 ([118]) If 𝐺 is a graph with diam(𝐺) ≥ 3, then 𝛾(𝐺) ≤ 2. In 2010 DeLaViña et al. [222] proved the following bound on the domination number in terms of the radius.

Section 4.5. Domination and Distance Theorem 4.39 ([222]) If 𝐺 is a connected graph, then 𝛾(𝐺) ≥

89 2 3

rad(𝐺).

The bound of Theorem 4.39 is tight. For example, if 𝐺 = 𝐶6𝑘 for 𝑘 ≥ 1, then rad(𝐺) = 3𝑘 and 𝛾(𝐺) = 2𝑘, that is, 𝛾(𝐺) = 23 rad(𝐺). On the other hand, for 𝑘 ≥ 1 if 𝐺 = 𝑃2𝑘+1 ◦ 𝐾1 , then rad(𝐺) = 𝑘 + 1 and 𝛾(𝐺) = 2𝑘 + 1. In this case, 𝛾(𝐺) = 2 rad(𝐺) − 1, showing that the difference between these two values can be arbitrarily large. In 2007 DeLaViña et al. [221] proved that the radius of a connected graph is a lower bound on its total domination number. In order to prove this result, they first proved the following three preliminary lemmas. Lemma 4.40 ([221]) If 𝐷 is a dominating set in a tree 𝑇, then the subgraph 𝑇 − 𝐷 has at most 𝑘 − 1 edges, where 𝑘 is the number of components of the subgraph 𝑇 [𝐷] induced by 𝐷. Proof Let 𝑇 = (𝑉, 𝐸) be a tree of order 𝑛 and size 𝑚. Let 𝐷 be a dominating set in 𝑇 and let 𝑘 be the number of components of the subgraph 𝑇 [𝐷]. We note that the subgraph 𝑇 − 𝐷 is the forest 𝑇 [𝑉 \ 𝐷] induced by 𝑉 \ 𝐷. Let 𝑚 1 denote the number of edges in 𝑇 [𝐷], let 𝑚 2 be the number of edges between 𝐷 and 𝑉 \ 𝐷, and let 𝑚 3 denote the number of edges in 𝑇 [𝑉 \ 𝐷]. Since 𝑇 is a tree, 𝑛 − 1 = 𝑚 = 𝑚 1 + 𝑚 2 + 𝑚 3 . We wish to show that 𝑚 3 ≤ 𝑘 − 1. Suppose, to the contrary, that 𝑚 3 ≥ 𝑘. Since 𝑇 [𝐷] is a forest with 𝑘 (tree) components, 𝑚 1 = |𝐷| − 𝑘. Since 𝐷 is a dominating set, every vertex in 𝑉 \ 𝐷 is adjacent to at least one vertex
in 𝐷, and so
𝑚 2 ≥ |𝑉 \ 𝐷| = 𝑛 − |𝐷|. Therefore, 𝑛 − 1 = 𝑚 1 + 𝑚 2 + 𝑚 3 ≥ |𝐷 | − 𝑘 + 𝑛 − |𝐷 | + 𝑘 = 𝑛, a contradiction. Consequently, 𝑚 3 ≤ 𝑘 − 1. Lemma 4.41 ([221]) If 𝐷 is a 𝛾t -set in a nontrivial connected graph 𝐺, then there exists a spanning tree 𝑇 of 𝐺 such that the following properties hold: (a) 𝐷 is a 𝛾t -set in 𝑇, and (b) the number of components in 𝑇 [𝐷] equals the number of components in 𝐺 [𝐷]. Proof Let 𝐺 be a nontrivial connected graph. Let 𝐷 be a 𝛾t -set of 𝐺, and let 𝐷 = 𝑉 \ 𝐷. If 𝐺 is a tree, then we take 𝑇 = 𝐺, and the desired result is immediate. Otherwise, 𝐺 contains at least one cycle 𝐶. We delete an edge 𝑒 from 𝐶 forming 𝐺 𝑒 = 𝐺 − 𝑒 as follows. If 𝐶 has two consecutive vertices 𝑢 and 𝑣 in 𝐷, then let 𝑒 = 𝑢𝑣 and delete the edge 𝑒. We note that in this case, the set 𝐷 remains a TD-set of the (connected) graph 𝐺 𝑒 . Further, the number of components in 𝐺 𝑒 [𝐷] equals the number of components in 𝐺 [𝐷]. If 𝐶 has no two consecutive vertices in 𝐷, but 𝐶 has two consecutive vertices 𝑢 and 𝑣 such that 𝑢 ∈ 𝐷 and 𝑣 ∈ 𝐷, then let 𝑒 = 𝑢𝑣 and delete the edge 𝑒. We note that in this case, the second neighbor of 𝑣 on 𝐶 belongs to 𝐷, and therefore the set 𝐷 is a TD-set of the (connected) graph 𝐺 𝑒 . Further, the number of components in 𝐺 𝑒 [𝐷] equals the number of components in 𝐺 [𝐷]. If neither of the above two cases applies, then the cycle 𝐶 is entirely contained in 𝐺 [𝐷]. In this case, we choose 𝑒 as an arbitrary edge of 𝐶 and delete the edge 𝑒. Once again, the set 𝐷 is still a TD-set of the (connected) graph 𝐺 𝑒 , and the number of components in 𝐺 𝑒 [𝐷] equals the number of components in 𝐺 [𝐷].

Chapter 4. General Bounds

90

We now repeat the above process in the connected graph 𝐺 𝑒 until all cycles are removed. Let 𝑇 denote the resulting subgraph. Since we are careful always to preserve connectivity, 𝑇 is a spanning tree of 𝐺. By construction, 𝐷 is a TD-set of 𝑇, and so 𝛾t (𝑇) ≤ |𝐷| = 𝛾t (𝐺). By Observation 4.2, 𝛾t (𝐺) ≤ 𝛾t (𝑇). Consequently, 𝛾t (𝑇) = 𝛾t (𝐺) and 𝐷 is a 𝛾t -set in 𝑇. Lemma 4.42 ([221]) If 𝐺 is a nontrivial connected graph, then 𝛾t (𝐺) ≥

1 2


diam(𝐺) + 1 .

Proof Let 𝐷 be a 𝛾t -set in 𝐺, and let 𝑘 be the number of components in 𝐺 [𝐷]. By Lemma 4.41, there exists a spanning tree 𝑇 of 𝐺 such that 𝐷 is a 𝛾t -set in 𝑇 and the number of components in 𝑇 [𝐷] equals 𝑘. Since 𝐷 is a TD-set of 𝑇, every component in 𝑇 [𝐷] contains at least two vertices. Thus, since 𝑇 [𝐷] is a forest with 𝑘 (tree) components, 𝛾t (𝑇) = |𝐷 | ≥ 2𝑘. Let 𝑃 be a longest path in 𝑇, and so |𝐸 (𝑃)| = diam(𝑇). Let 𝑝 1 denote the number of edges of 𝑃 in 𝑇 [𝐷], let 𝑝 2 be the number of edges of 𝑃 between 𝐷 and 𝑉 \ 𝐷, and let 𝑝 3 denote the number of edges of 𝑃 in 𝑇 [𝑉 \ 𝐷]. Since 𝑇 [𝐷] is a forest with 𝑘 components, there are |𝐷| − 𝑘 = 𝛾t (𝑇) − 𝑘 edges in 𝑇 [𝐷], and so 𝑝 1 ≤ 𝛾t (𝑇) − 𝑘. In traversing a longest path in 𝑇, we can enter and leave each component of 𝑇 [𝐷] at most once, implying that 𝑝 2 ≤ 2𝑘. By Lemma 4.40, the subgraph 𝑇 − 𝐷 has at most 𝑘
− 1 edges, and so 𝑝 3 ≤ 𝑘 − 1. Thus, diam(𝑇) = |𝐸 (𝑃)| = 𝑝 1 + 𝑝 2 + 𝑝 3 ≤ 𝛾t (𝑇) − 𝑘 + 2𝑘 + (𝑘 − 1) = 𝛾t (𝑇) + 2𝑘 − 1 ≤ 2𝛾t (𝑇) − 1. However, since the diameter of a graph is at most the diameter of any of its spanning trees, diam(𝐺) ≤ diam(𝑇) ≤ 2𝛾t (𝑇) − 1, or
equivalently, 𝛾t (𝐺) ≥ 12 diam(𝐺) + 1 . With the above three lemmas, DeLaViña et al. [221] proved that the radius of a connected graph is a lower bound on its total domination number. Theorem 4.43 ([221]) If 𝐺 is a nontrivial connected graph, then 𝛾t (𝐺) ≥ rad(𝐺). Proof Let 𝐷 be a 𝛾t -set in 𝐺. By Lemma 4.41, there exists a spanning tree 𝑇 of 𝐺 such that 𝐷 is a 𝛾t -set in 𝑇. Since 𝑇 is a tree, 2 rad(𝑇) − 1 ≤ diam(𝑇). By Lemma 4.42, diam(𝑇) ≤ 2𝛾t (𝑇) − 1 = 2|𝐷 | − 1 = 2𝛾t (𝐺) − 1. Consequently, 2 rad(𝑇) − 1 ≤ 2𝛾t (𝐺) − 1, or equivalently, rad(𝑇) ≤ 𝛾t (𝐺). Since the radius of a graph is at most the radius of any of its spanning trees, rad(𝐺) ≤ rad(𝑇). Therefore, rad(𝐺) ≤ 𝛾t (𝐺). We note that if 𝐺 = 𝑃4𝑘 where 𝑘 ≥ 1, then 𝛾t (𝐺) = 2𝑘 = rad(𝐺). Furthermore, in this case, the graph 𝐺 has a unique 𝛾t -set 𝐷 and the subgraph 𝐺 [𝐷] = 𝑘𝐾2 , where 𝑘 = 12 rad(𝐺). DeLaViña et al. [221] characterized the graphs attaining the bound of Theorem 4.43. Theorem 4.44 ([221]) If 𝐷 is a 𝛾t -set of a nontrivial
connected graph 𝐺, then 𝛾t (𝐺) = rad(𝐺) if and only if 𝐺 [𝐷] has size 𝑚 𝐺 [𝐷] = 12 rad(𝐺).

Section 4.6. Domination and Packing

91

Proof Suppose that 𝛾t (𝐺) = rad(𝐺). Let 𝐷 be a 𝛾t -set in 𝐺, and let 𝑘 be the number of components in 𝐺 [𝐷]. By Lemma 4.41, there exists a spanning tree 𝑇 of 𝐺 such that 𝐷 is a 𝛾t -set in 𝑇 and the number of components in 𝑇 [𝐷] equals 𝑘. As shown in the proofs of Lemma 4.42 and Theorem 4.43, and by our supposition that 𝛾t (𝐺) = rad(𝐺), the following inequality chain holds: =rad(𝐺) z}|{ (4.4) 2 rad(𝐺) − 1 ≤ diam(𝑇) ≤ 𝛾t (𝑇) + 2𝑘 − 1 ≤ 2𝛾t (𝑇) − 1 = 2𝛾t (𝐺) − 1. We must have equality throughout Inequality Chain (4.4), implying that rad(𝐺) = 2𝑘 = 𝛾(𝐺). Since 𝐷 is a TD-set of 𝐺, every component in 𝐺 [𝐷] contains at least two vertices, and so 2𝑘 = 𝛾t (𝐺) = |𝐷 | ≥ 2𝑘. Consequently, |𝐷| = 2𝑘, implying that each of the 𝑘 components of 𝐺 [𝐷] is a
𝐾2 -component. Therefore, 𝐺 [𝐷] has size equal to 𝑘 = 21 rad(𝐺), that is, 𝑚 𝐺 [𝐷] = 12 rad(𝐺).
Conversely, suppose that 𝑚 𝐺 [𝐷] = 12 rad(𝐺). Since 𝐷 is a TD-set of 𝐺, every vertex in 𝐷 has degree at least 1 in 𝐺 [𝐷], and so, by Theorem 4.43, the following inequality chain holds:
∑︁ rad(𝐺) = 2𝑚 𝐺 [𝐷] = deg𝐺 [𝐷 ] (𝑣) ≥ |𝐷| = 𝛾t (𝐺) ≥ rad(𝐺) (4.5) 𝑣 ∈𝐷

We must have equality throughout Inequality (4.5), implying that 𝛾t (𝐺) = rad(𝐺).

4.6

Domination and Packing

Recall that 𝜌(𝐺) and 𝜌 o (𝐺) denote the packing number and open packing number, respectively, of 𝐺. The following result shows that the domination number of a graph is at least its packing number. Theorem 4.45 If 𝐺 is a graph, then 𝛾(𝐺) ≥ 𝜌(𝐺). Proof Let 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } be a maximum packing in 𝐺, and so 𝑘 = 𝜌(𝐺). Let 𝐷 be a 𝛾-set in 𝐺. In order to dominate the vertex 𝑣 𝑖 , we have |𝐷 ∩ N[𝑣 𝑖 ] | ≥ 1 for all 𝑖 ∈ [𝑘]. Since N[𝑣 1 ], N[𝑣 2 ], . . . , N[𝑣 𝑘 ] are vertex-disjoint sets, 𝛾(𝐺) = |𝐷| ≥

𝑘 ∑︁

|𝐷 ∩ N[𝑣 𝑖 ] | ≥ 𝑘 = 𝜌(𝐺).

𝑖=1

A result similar to that of Theorem 4.45 holds for total domination and open packing numbers. Theorem 4.46 If 𝐺 is an isolate-free graph, then 𝛾t (𝐺) ≥ 𝜌 o (𝐺). Proof Let 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } be a maximum open packing in 𝐺, and so 𝑘 = 𝜌 o (𝐺). Let 𝐷 be a 𝛾t -set in 𝐺. In order to totally dominate the vertex 𝑣 𝑖 , we have

Chapter 4. General Bounds

92

|𝐷∩N(𝑣 𝑖 )| ≥ 1 for all 𝑖 ∈ [𝑘]. Since N(𝑣 1 ), N(𝑣 2 ), . . . , N(𝑣 𝑘 ) are vertex-disjoint sets, 𝛾t (𝐺) = |𝐷 | ≥

𝑘 ∑︁

|𝐷 ∩ N(𝑣 𝑖 )| ≥ 𝑘 = 𝜌 o (𝐺).

𝑖=1

In 1975 Meir and Moon [589] proved that the domination number of a tree equals its packing number. Similarly, in 2005 Rall [644] proved that the total domination number of a tree equals its open packing number. Proofs of these results will be given in Chapter 5.

4.7

Gallai Type Theorems

As we mentioned in Chapter 1, after Gallai [324] published his well-known theorem, that for any connected graph 𝐺 of order 𝑛, the sum of the independence number of 𝐺 plus the vertex covering number of 𝐺 always equals the order 𝑛 of 𝐺, a number of other theorems of this form, A (𝐺) + B (𝐺) = 𝑛, began to be published. Results like this are often called Gallai theorems. For all six of the core domination parameters in this book, there are corresponding Gallai theorems. Theorem 4.47 For any connected graph 𝐺 of order 𝑛 ≥ 2, the following hold: (a) 𝛼(𝐺) + 𝛽(𝐺) = 𝑛 (1959 Gallai [324]). (b) 𝛾(𝐺) + 𝜀 𝑓 (𝐺) = 𝑛 (1974 Nieminen [613]). (c) 𝛾(𝐺) + Ψ(𝐺) = 𝑛 (1977 Slater [679]). (d) Γ(𝐺) + 𝜓(𝐺) = 𝑛 (1977 Slater [679]). (e) 𝑖(𝐺) + 𝛽+ (𝐺) = 𝑛 (1980 McFall and Nowakowski [587]). (f) 𝛾t (𝐺) + Ψo (𝐺) = 𝑛 (1988 Cockayne et al. [195]). (g) Γt (𝐺) + 𝜓 o (𝐺) = 𝑛 (1988 Cockayne et al. [195]). As a reminder, other than the six core parameters which appear on the left in these equalities, the corresponding parameters in these sums are the following: (a) 𝛽(𝐺), the minimum cardinality of a vertex cover of 𝐺. (b) 𝜀 𝑓 (𝐺), the maximum number of pendant edges in a spanning forest of 𝐺. (c) Ψ(𝐺), the maximum cardinality of an enclaveless set in 𝐺. (d) 𝜓(𝐺), the minimum cardinality of a maximal enclaveless set in 𝐺. (e) 𝛽+ (𝐺), the maximum cardinality of a minimal vertex cover of 𝐺. (f) Ψo (𝐺), the maximum cardinality of an open enclaveless set in 𝐺. (g) 𝜓 o (𝐺), the minimum cardinality of an open enclaveless set in 𝐺. From these equalities above and other established inequalities, a number of inequalities can be established where the sum of two parameters is bounded above by 𝑛. For example, since 𝑖(𝐺) ≤ 𝛼(𝐺) and 𝛾(𝐺) ≤ 𝛽(𝐺) if 𝐺 is isolate-free, we can conclude from Gallai’s Theorem that 𝑖(𝐺) + 𝛾(𝐺) ≤ 𝑛. But this inequality can be improved, as was shown by Allan et al. [16] in 1984, as follows. Theorem 4.48 ([16]) If 𝐺 is a connected graph of order 𝑛 ≥ 3, then 𝑖(𝐺) + 𝛾t (𝐺) ≤ 𝑛. Proof Among all 𝛾-sets of 𝐺, let 𝑆 be chosen so that

Section 4.7. Gallai Type Theorems

93

(a) the number of isolated vertices in 𝐺 [𝑆] is a minimum, and (b) subject to (a), the sum of degrees of vertices that belong to 𝑆 is a maximum. Let 𝑆 ′ ⊆ 𝑆 be the set of isolated vertices in 𝐺 [𝑆] and assume that |𝑆 ′ | = 𝑘. If 𝑘 = 0, then 𝑆 is a TD-set and therefore 𝛾(𝐺) = 𝛾t (𝐺), and since 𝑖(𝐺) + 𝛾(𝐺) ≤ 𝑛 it follows that 𝑖(𝐺) + 𝛾t (𝐺) ≤ 𝑛, as required. Assume therefore that 𝑘 > 0. Suppose that deg(𝑣) = 1 for some 𝑣 ∈ 𝑆 ′ . Let 𝑢 be the only neighbor of 𝑣. By supposition, we
have 𝑛 ≥ 3, implying that deg(𝑢) ≥ 2. We now consider the 𝛾-set 𝑆 ′′ = 𝑆 \ {𝑣} ∪ {𝑢}. If 𝑢 is not isolated in 𝐺 [𝑆 ′′ ], then 𝑆 ′′ is a 𝛾-set having fewer isolated vertices in 𝐺 [𝑆 ′′ ] than in 𝐺 [𝑆], contradicting our choice of 𝑆. Hence, 𝑢 is isolated in 𝐺 [𝑆 ′′ ]. However, the sum of degrees of vertices that belong to 𝑆 ′′ exceeds that in 𝑆, once again contradicting our choice of 𝑆. Hence, deg(𝑣) ≥ 2 for all 𝑣 ∈ 𝑆 ′ . We claim that each 𝑣 ∈ 𝑆 has an 𝑆-external private neighbor, that is, |epn[𝑣, 𝑆] | ≥ 1 for all 𝑣 ∈ 𝑆. By Ore’s Lemma 2.72, we have ipn[𝑣, 𝑆] ≠ ∅ or epn[𝑣, 𝑆] ≠ ∅. If 𝑣 ∈ 𝑆 \ 𝑆 ′ , then ipn[𝑣, 𝑆] = ∅, and therefore epn[𝑣, 𝑆] ≠ ∅, as desired. Let 𝑣 ∈ 𝑆 ′ and suppose, to the contrary, that epn[𝑣, 𝑆] = ∅. Thus, every neighbor of 𝑣 (which necessarily belongs to the set 𝑉 \ 𝑆) has at least two neighbors in 𝑆. In this case, replacing 𝑣 in the set 𝑆 with an arbitrary neighbor of 𝑣 produces a new 𝛾-set of 𝐺 having fewer isolated vertices in its induced subgraph than does 𝑆, contradicting our choice of 𝑆. Hence, epn[𝑣, 𝑆] ≠ ∅. Thus, |epn[𝑣, 𝑆] | ≥ 1 for all 𝑣 ∈ 𝑆. For each vertex 𝑣 ∈ 𝑆, let 𝑤 𝑣 be an arbitrary vertex in epn[𝑣, 𝑆]. By our earlier observations, deg(𝑣) ≥ 2 for all 𝑣 ∈ 𝑆 ′ . For each vertex 𝑣 ∈ 𝑆 ′ , let 𝑢 𝑣 be a neighbor of 𝑣 different from 𝑤 𝑣 . Let Ø Ø 𝑈= {𝑤 𝑣 } and 𝑊= {𝑢 𝑣 }. 𝑣 ∈𝑆 ′

𝑣 ∈𝑆

We note that |𝑊 | = |𝑆| = 𝛾(𝐺). We also note that |𝑈| ≤ |𝑆 ′ | and 𝑈 ⊆ 𝑉 \ (𝑆 ∪𝑊). Moreover, the set 𝑆 ∪ 𝑈 is a TD-set of 𝐺. Therefore, 𝛾t (𝐺) ≤ |𝑆| + |𝑈| = 𝛾(𝐺) + |𝑈|.

(4.6)

𝑆′

We can extend the independent set to a maximal independent set of 𝐺, call it 𝑋. Therefore, 𝑖(𝐺) ≤ |𝑋 |. For each vertex 𝑣 ∈ 𝑆, at most one of two adjacent vertices 𝑣 and 𝑤 𝑣 belongs to the set 𝑋. Therefore, there exists a subset 𝑊 ′ ⊂ 𝑆 ∪ 𝑊 such that |𝑊 ′ | = 𝛾(𝐺) and 𝑊 ′ ∩ 𝑋 = ∅. In addition, since 𝑆 ′ ⊆ 𝑋 and each vertex in 𝑆 ′ has a neighbor in 𝑈, we note that 𝑈 ∩ 𝑋 = ∅. Hence, by our earlier observations, the sets 𝑈, 𝑋, and 𝑊 ′ are pairwise disjoint, and so |𝑈| + |𝑋 | + |𝑊 ′ | = |𝑈 ∪ 𝑋 ∪ 𝑊 ′ | ≤ |𝑉 | = 𝑛. Since 𝑖(𝐺) ≤ |𝑋 |, then by Inequality (4.6), 𝑖(𝐺) + 𝛾t (𝐺) ≤ |𝑋 | + 𝛾(𝐺) + |𝑈| = |𝑋 | + |𝑊 ′ | + |𝑈| = |𝑈 ∪ 𝑋 ∪ 𝑊 ′ | ≤ 𝑛. If 𝐺 = 𝑘𝐾3 for some 𝑘 ≥ 1, then 𝑖(𝐺) = 𝑘 and 𝛾t (𝐺) = 2𝑘, while 𝑛 = 3𝑘. Thus, in this case, 𝑖(𝐺) + 𝛾t (𝐺) = 𝑛. Hence, the bound in Theorem 4.48 is best possible. We remark that if 𝐻 = 𝑘𝐾2 for some 𝑘 ≥ 1, then 𝑖(𝐺) = 𝑘 and 𝛾t (𝐺) = 2𝑘, while 𝑛 = 2𝑘. Thus, in this case, 𝑖(𝐺) + 𝛾t (𝐺) = 32 𝑛. Hence, the requirement in this theorem that every component of 𝐺 has order at least 3 is necessary.

94

Chapter 4. General Bounds

4.8 Domination and Matching Recall that the matching number 𝛼′ (𝐺) of 𝐺 is the maximum cardinality of a matching in 𝐺. As an immediate consequence of Lemma 4.22 by Bollobás and Cockayne, the domination number of an isolate-free graph is at most its matching number. Corollary 4.49 ([84]) If 𝐺 is an isolate-free graph, then 𝛾(𝐺) ≤ 𝛼′ (𝐺). Proof By Lemma 4.22, the graph 𝐺 contains a 𝛾-set 𝐷 such that epn[𝑣, 𝐷] ≠ ∅ for every vertex 𝑣 ∈ 𝐷. For each vertex 𝑣 ∈ 𝐷, let 𝑣 ′ be an arbitrary vertex in epn[𝑣, 𝐷]. The set 𝑀 = {𝑣𝑣 ′ : 𝑣 ∈ 𝐷} is a matching in 𝐺, implying that 𝛼′ (𝐺) ≥ |𝑀 | = |𝐷| = 𝛾(𝐺). We note that for a corona 𝐺 = 𝐻 ◦ 𝐾1 of any graph 𝐻, 𝛾(𝐺) = 𝛼′ (𝐺). Since the vertices incident to the edges in a maximum matching in a graph form a TD-set in the graph, the total domination number of an isolate-free graph is trivially at most twice the matching number. Observation 4.50 If 𝐺 is an isolate-free graph, then 𝛾t (𝐺) ≤ 2𝛼′ (𝐺). Since 𝛾(𝐺) ≤ 𝛾t (𝐺) for all isolate-free graphs 𝐺, it is natural to ask whether the total domination number lies between the domination number and the matching number. However, it is not necessarily true that 𝛾t (𝐺) ≤ 𝛼′ (𝐺) for all isolate-free graphs 𝐺. For example, if 𝐺 = 𝑃2𝑘 ◦ 𝑃2 is the 2-corona of a path 𝑃2𝑘 for 𝑘 ≥ 1, then 𝐺 has order 𝑛 = 6𝑘, 𝛼′ (𝐺) = 12 𝑛 = 3𝑘, and 𝛾t (𝐺) = 23 𝑛 = 4𝑘, showing that the difference 𝛾t (𝐺) − 𝛼′ (𝐺) = 𝑘 can be arbitrarily large. It is also not necessarily true that 𝛼′ (𝐺) ≤ 𝛾t (𝐺) for all isolate-free graphs 𝐺. As a simple example, if 𝐺 is the graph obtained from a complete graph 𝐾2𝑘+5 for 𝑘 ≥ 1 by adding a new vertex (of degree 1) and joining it to one vertex of the complete graph, then 𝐺 has order 𝑛 = 2(𝑘 + 3), 𝛼′ (𝐺) = 12 𝑛 = 𝑘 + 3, and 𝛾t (𝐺) = 2, showing that the difference 𝛼′ (𝐺) − 𝛾t (𝐺) can be arbitrarily large. More generally, Henning et al. [465] in 2008 showed that even for graphs 𝐺 with arbitrarily large minimum degree (but fixed with respect to the order of the graph), it is not necessarily true that the inequality 𝛾t (𝐺) ≤ 𝛼′ (𝐺) must hold, or that the inequality 𝛼′ (𝐺) ≤ 𝛾t (𝐺) must hold. Theorem 4.51 ([465]) For every integer 𝛿 ≥ 1, there exist graphs 𝐺 and 𝐻 with 𝛿(𝐺) = 𝛿(𝐻) = 𝛿 satisfying 𝛾t (𝐺) > 𝛼′ (𝐺) and 𝛾t (𝐻) < 𝛼′ (𝐻). A path covering of a graph 𝐺 is a collection of vertex-disjoint paths of 𝐺 that partition 𝑉 (𝐺). The minimum cardinality of a path covering of 𝐺 is the path covering number of 𝐺, denoted pc(𝐺). DeLaViña et al. [221] established a relationship between the total domination number, the matching number, and the path covering number of an isolate-free graph. Theorem 4.52 ([221]) If 𝐺 is a nontrivial connected graph, then 𝛾t (𝐺) ≤ 𝛼′ (𝐺) + pc(𝐺). Proof Let Q be a path covering of 𝐺 of minimum cardinality, that is, |Q| = pc(𝐺) and every vertex of 𝐺 belongs to exactly one path in Q. Let Q2 ⊆ Q be the paths

Section 4.8. Domination and Matching

95

in Q of order at least 2. Let  𝑃 be an arbitrary path in Q2 , and let 𝑃 have order 𝑝 ≥ 2. We note that 𝛼′ (𝑃) ≥ 𝑝2 and, by Proposition 4.11, 𝛾t (𝑃) =

 𝑝 2

+

 𝑝 4

−

 𝑝 4

≤

 𝑝 2

+ 1 ≤ 𝛼′ (𝑃) + 1.

Taking the union of a maximum matching over all paths 𝑃 ∈ Q2 produces a matching in 𝐺. Therefore, by our earlier observations,  ∑︁  ∑︁ ∑︁
𝛾t (𝑃) ≤ 𝛼′ (𝑃) + 1 = 𝛼′ (𝑃) + |Q2 | ≤ 𝛼′ (𝐺) + |Q2 |. (4.7) 𝑃 ∈ Q2

𝑃∈ Q2

𝑃∈ Q2

Let Q1 be the paths in Q of order 1. Possibly, Q1 = ∅. However, if Q1 ≠ ∅, then by the minimality of the path covering Q, no two vertices in different paths in Q1 are adjacent. For each path 𝑃 ∈ Q2 , let 𝑆 𝑃 be a 𝛾t -set of the path 𝑃. For each path 𝑃 ∈ Q1 , let 𝑣 𝑃 be the one vertex on the path 𝑃, and let 𝑥 𝑃 be an arbitrary neighbor of 𝑣 𝑃 . By our earlier observations, the vertex 𝑥 𝑃 lies on a path 𝑄 that belongs to Q2 and therefore the vertex 𝑥 𝑃 is totally dominated by the set 𝑆 𝑄 . Let Ø  Ø  𝑆= 𝑆𝑃 ∪ {𝑥 𝑃 } . 𝑃∈ Q2

𝑃∈ Q1

The set 𝑆 so constructed is a TD-set of 𝐺. Hence, by Inequality (4.7), ∑︁ ∑︁ 𝛾t (𝐺) ≤ |𝑆| ≤ |𝑆 𝑃 | + |{𝑥 𝑃 }| 𝑃∈ Q2

=

∑︁

𝑃∈ Q1

𝛾t (𝑃) + |Q1 |

𝑃∈ Q2 ′


≤ 𝛼 (𝐺) + |Q2 | + |Q1 | = 𝛼′ (𝐺) + |Q| = 𝛼′ (𝐺) + pc(𝐺). That the bound of Theorem 4.52 is tight may be seen as follows. For 𝑘 ≥ 1, let 𝐺 𝑘 be the graph of order 7𝑘 obtained from a cycle 𝐶 𝑘 , if 𝑘 ≥ 3, or a path 𝑃 𝑘 , if 𝑘 = 1 or 𝑘 = 2, by identifying each vertex of the cycle or path with the center of a path 𝑃7 . The resulting graph 𝐺 𝑘 satisfies 𝛾t (𝐺 𝑘 ) = 4𝑘, 𝛼′ (𝐺 𝑘 ) = 3𝑘, and pc(𝐺 𝑘 ) = 𝑘, that is, 𝛾t (𝐺 𝑘 ) = 𝛼′ (𝐺 𝑘 ) + pc(𝐺 𝑘 ). When 𝑘 = 4, the graph 𝐺 𝑘 is illustrated in Figure 4.4, where the highlighted vertices form a 𝛾t -set of 𝐺 𝑘 . In view of Theorem 4.51, a natural problem is to determine for which graph classes G ′ it is true that 𝛾t (𝐺) ≤ 𝛼′ (𝐺) for all graphs 𝐺 ∈ G ′ . Several such classes of graphs are given in the literature. We list only a few of these given by Henning et al. [465] and Henning and Yeo [480, 489].

Chapter 4. General Bounds

96

Figure 4.4 The graph 𝐺 4

Theorem 4.53 If 𝐺 is a connected graph of order 𝑛, then the following hold: (a) ([465]) If 𝐺 is claw-free with 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 𝛼′ (𝐺). (b) ([480]) If 𝐺 is 𝑟-regular where 𝑟 ≥ 3, then 𝛾t (𝐺) ≤ 𝛼′ (𝐺). (c) ([489]) If all vertices of 𝐺 are contained in a triangle and 𝑛 ≥ 4, then 𝛾t (𝐺) ≤ 𝛼′ (𝐺). We omit proofs of the results in Theorem 4.53(a) and (c). However, we do present a proof of the 3-regular case in Theorem 4.53(b) since as remarked by O and West [617], the “3-regular case is the only case where the inequality between 𝛾t and 𝛼′ is delicate. When more edges are added, 𝛼′ tends to increase and 𝛾t tends to decrease, so the separation increases.” For this purpose, we recall a classic result about the matching number of a graph due to Berge [66], which is sometimes referred to as the Tutte-Berge formulation for the matching number. For a graph 𝐺 and a subset 𝑋 ⊂ 𝑉 (𝐺), let oc(𝐺 − 𝑋) denote the number of components of 𝐺 − 𝑋 that have odd order. Theorem 4.54 (Tutte-Berge Formula [66]) For every graph 𝐺 and subset 𝑋 ⊆ 𝑉 (𝐺),
𝛼′ (𝐺) = min 12 |𝑉 (𝐺)| + |𝑋 | − oc(𝐺 − 𝑋) . 𝑋⊆𝑉 (𝐺)

We shall also need the following result that is proved in Chapter 8. Theorem 4.55 ([458, 672]) If 𝐺 is a connected graph of order
𝑛 ≥ 3 and size 𝑚 with Δ(𝐺) ≤ Δ for some integer Δ ≥ 3, then 𝑚 ≤ Δ 𝑛 − 𝛾t (𝐺) . Theorem 4.56 ([480]) If 𝐺 is a 3-regular connected graph, then 𝛾t (𝐺) ≤ 𝛼′ (𝐺). Proof Let 𝐺 be a 3-regular connected graph of order 𝑛. By the Tutte-Berge Formula in Theorem 4.54, there is a subset 𝑋 of vertices in 𝐺 satisfying
𝛼′ (𝐺) = 12 |𝑉 (𝐺)| + |𝑋 | − oc(𝐺 − 𝑋) . (4.8) If 𝑋 = ∅, then oc(𝐺 − 𝑋) = oc(𝐺) = 0, and so by Equation (4.8), we have 𝛼′ (𝐺) = 12 𝑛. Applying Theorem 4.55 with Δ = 3 to the 3-regular graph 𝐺, we have

Section 4.8. Domination and Matching

97


= 𝑚 ≤ 3 𝑛 − 𝛾t (𝐺) , or equivalently, 𝛾t (𝐺) ≤ 12 𝑛 = 𝛼′ (𝐺). Hence, we may assume that 𝑋 ≠ ∅. Let 𝐺 ★ be an arbitrary odd component of 𝐺 − 𝑋, and let 𝐺 ★ have (odd) order 𝑛★ and size 𝑚★. If there are 𝑠 edges between 𝑋 and 𝐺 ★, then 2𝑚★ = 3𝑛★ − 𝑠, which implies that 𝑠 is odd since 𝑛★ is odd. Thus, every odd component of 𝐺 − 𝑋 is joined to 𝑋 by either exactly one edge or by at least three edges. Hence, if 𝑐 1 is the number of odd components of 𝐺 − 𝑋 having only one edge to 𝑋, then
counting the edges joining 𝑋 to odd components yields at least 𝑐 1 + 3 oc(𝐺 − 𝑋) − 𝑐 1 . However, since 𝐺 is 3-regular, the number of edges joining 𝑋 to odd components is at most 3|𝑋 |. Consequently, 3oc(𝐺 − 𝑋) − 2𝑐 1 ≤ 3|𝑋 |, or equivalently, oc(𝐺 − 𝑋) − |𝑋 | ≤ 23 𝑐 1 .

Hence, by Equation (4.8), we have 𝛼′ (𝐺) = 12 𝑛 + |𝑋 | − oc(𝐺 − 𝑋) ≥ 12 𝑛 − 23 𝑐 1 , yielding the inequality 𝛼′ (𝐺) ≥ 12 𝑛 − 13 𝑐 1 . (4.9) 3 2𝑛

We define a 1-odd-component of 𝐺 − 𝑋 as an odd component of 𝐺 − 𝑋 joined to 𝑋 by exactly one edge. Let 𝐺 ★ be a 1-odd-component of 𝐺 − 𝑋, and let 𝐺 ★ have (odd) order 𝑛★ and size 𝑚★. We note that one vertex of 𝐺 ★ has degree 2 in 𝐺 ★ and the other vertices of 𝐺 ★ have degree 3 in 𝐺 ★, and so 𝑚★ = 12 (3𝑛★ − 1). By Theorem 4.55, 𝛾t (𝐺 ★) ≤ 𝑛★ − 13 𝑚★ = 𝑛★ − 16 (3𝑛★ − 1) = 12 𝑛★ + 16 . Since 𝑛★ is odd and 𝛾t (𝐺 ★) is an integer, 𝛾t (𝐺 ★) ≤ 12 (𝑛★ − 1). Let 𝐺 ′ be the graph obtained from 𝐺 by removing all vertices that belong to a 1-odd-component of 𝐺 − 𝑋. Suppose that there is an isolated vertex 𝑥 in 𝐺 ′ . Since 𝐺 is connected, the graph 𝐺 is then determined and consists of the vertex 𝑥 and three 1-odd-components of 𝐺 − 𝑋, with the vertex 𝑥 joined to one vertex from each such component. In particular, 𝑐 1 = 3. Let 𝐺 1 , 𝐺 2 , and 𝐺 3 denote these three 1-odd-components, and let 𝑣 𝑖 be the vertex in 𝐺 𝑖 adjacent to 𝑥 for 𝑖 ∈ [3]. Let 𝐺 𝑖 have order 𝑛𝑖 and 𝐷 𝑖 be a 𝛾t -set of 𝐺 𝑖 for 𝑖 ∈ [3]. We note that 𝑛 = 𝑛1 + 𝑛2 + 𝑛3 + 1. By our earlier observations, |𝐷 𝑖 | ≤ 12 (𝑛𝑖 − 1). The set 𝐷 1 ∪ 𝐷 2 ∪ 𝐷 3 ∪ {𝑣 1 }, for example, is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ 12 (𝑛1 + 𝑛2 + 𝑛3 − 3) + 1 = 12 (𝑛 + 1 − 3) = 𝛼′ (𝐺) (noting here that |𝑋 | = 1 and oc(𝐺 − 𝑋) = 3). Hence, we may assume that 𝐺 ′ is isolate-free. Suppose that 𝐺 ′ has a component of order 2. Let {𝑥, 𝑦} be the vertex set of this component. Since 𝐺 is connected, the graph 𝐺 is then determined and consists of the (adjacent) vertices 𝑥 and 𝑦 and four 1-odd-components of 𝐺 − 𝑋 (where 𝑥 is joined to two such components and 𝑦 to the other two components). In particular, 𝑐 1 = 4. Let 𝐺 1 , 𝐺 2 , 𝐺 3 , and 𝐺 4 denote these four 1-odd-components. Adopting our earlier notation, let 𝐺 𝑖 have order 𝑛𝑖 , and let 𝐷 𝑖 be a 𝛾t -set of 𝐺 𝑖 for 𝑖 ∈ [4]. The set 𝐷 1 ∪𝐷 2 ∪𝐷 3 ∪𝐷 4 ∪{𝑥, 𝑦} is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ 12 (𝑛1 +𝑛2 +𝑛3 +𝑛4 −4)+2 = 1 ′ 2 (𝑛 + 2 − 4) = 𝛼 (𝐺) (noting here that |𝑋 | = 2 and oc(𝐺 − 𝑋) = 4). Hence, we may assume that every component in 𝐺 ′ has order at least 3. Let 𝐺 ′ have order 𝑛′ and size 𝑚 ′ . By assumption, 𝑛′ ≥ 3. Applying Theorem 4.55 with Δ = 3 to each component of 𝐺 ′ yields a TD-set 𝐷 ′ of 𝐺 ′ such that |𝐷 ′ | ≤ 𝑛′ − 𝑚 ′ /3. Since there are exactly 𝑐 1 edges joining 𝑉 (𝐺 ′ ) and 𝑉 (𝐺) \ 𝑉 (𝐺 ′ ), ∑︁ 2𝑚 ′ = deg𝐺 ′ (𝑣) = 3𝑛′ − 𝑐 1 . (4.10) 𝑣 ∈𝑉 (𝐺 ′ )

Chapter 4. General Bounds

98

Let 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑐1 denote the 1-odd-components in 𝐺 − 𝑋. Adopting our earlier notation, let 𝐺 𝑖 have order 𝑛𝑖 and let 𝐷 𝑖 be a 𝛾t -set of 𝐺 𝑖 for 𝑖 ∈ [𝑐 1 ]. By our earlier observations, |𝐷 𝑖 | ≤ 21 (𝑛𝑖 − 1) for 𝑖 ∈ [𝑐 1 ]. We note that 𝑛 = 𝑛′ + 𝑛1 + · · · + 𝑛𝑐1 . Consider now the set 𝐷 = 𝐷 ′ ∪ 𝐷 1 ∪ · · · ∪ 𝐷 𝑐1 . The set 𝐷 is a TD-set of 𝐺. Thus, by Inequality (4.9) and Equation (4.10), and our earlier observations, 𝛾t (𝐺) ≤ |𝐷| = |𝐷 ′ | +

𝑐1 ∑︁

|𝐷 𝑖 |

𝑖=1

≤ 𝑛′ − 13 𝑚 ′ +

𝑐1 ∑︁

1 2 (𝑛𝑖

− 1)

𝑖=1

≤ 𝑛′ − 16 (3𝑛′ − 𝑐 1 ) − 12 𝑐 1 +

𝑐1 ∑︁

1 2 𝑛𝑖

𝑖=1

= 12 𝑛 − 13 𝑐 1 ≤ 𝛼′ (𝐺). O and West [617] defined a balloon in a graph 𝐺 as a maximal 2-edge-connected subgraph incident to exactly one cut-edge of 𝐺. As remarked by O and West, the “term arises from viewing the cut-edge as a string tying the balloon to the rest of the graph.” Further, they let 𝑏(𝐺) be the number of balloons in 𝐺. We remark that using our notation in the proof of Theorem 4.56, we have that 𝑏(𝐺) ≥ 𝑐 1 , and so Inequality (4.9) can be written as 𝛼′ (𝐺) ≥ 12 𝑛 − 13 𝑏(𝐺).

(4.11)

However, O and West [617] showed that going deeper into the proof of Theorem 4.56, one can improve the upper bound on the total domination number slightly. Theorem 4.57 ([617]) If 𝐺 is a 3-regular connected graph of order 𝑛, then 𝛾t (𝐺) ≤ 12 𝑛 − 12 𝑏(𝐺),

(4.12)

(except that 𝛾t (𝐺) ≤ 12 𝑛 − 1 when 𝑏(𝐺) = 3 and the three balloons have a common neighbor), and this is tight for all even values of 𝑏(𝐺). Inequalities (4.11) and (4.12) together improve the bound 𝛾t (𝐺) ≤ 𝛼′ (𝐺) for 3-regular connected graphs. Corollary 4.58 ([617]) If 𝐺 is a 3-regular connected graph of order 𝑛, then 𝛾t (𝐺) ≤ 𝛼′ (𝐺) − 16 𝑏(𝐺), except when 𝑏(𝐺) = 3 and there is exactly one vertex outside the balloons, in which case 𝛾t (𝐺) ≤ 𝛼′ (𝐺).

Chapter 5

Domination in Trees 5.1

Introduction

Trees are perhaps the simplest of all graph families, so it is not surprising that much research has been done involving domination in trees. Some of the most significant results on domination in trees are presented in this chapter. Many problems that are NP-complete for general graphs can be solved in polynomial time for trees. This is the case for the decision problem DOMINATING SET. In fact, linear algorithms exist for computing 𝛾(𝑇), 𝑖(𝑇), 𝛼(𝑇), Γ(𝑇), 𝛾t (𝑇), and Γt (𝑇) for arbitrary trees 𝑇, a discussion of which is found in Chapter 3. Interest in trees also arises because they are minimally connected, that is, among all connected graphs on 𝑛 vertices, trees have the minimum possible number, namely, 𝑛 − 1, of edges. This inherent skeleton-like structure of trees gives rise to many natural applications, ranging from representing data structures in theoretical computer science to modeling molecules in chemistry. Domination in trees aids in the study of some of these applications. For example, chemical indices of trees having a given domination number were investigated in [742] and in [755]. For a different type of example, domination numbers of trees were used in [434] to predict which trees were likely (or unlikely) to represent native RNA structures. In this chapter, we present theoretical results involving domination in trees. We introduce some additional terminology for this presentation. Recall that a vertex of degree 1 is a leaf, its neighbor is a support vertex, and a support vertex adjacent to two or more leaves is a strong support vertex. For an integer 𝑖 ≥ 1 and a tree 𝑇, the number of vertices of degree 𝑖 in 𝑇 is denoted by 𝑛𝑖 (𝑇), or simply by 𝑛𝑖 . Thus, 𝑛1 is the number of leaves of 𝑇. A caterpillar is a tree with the property that removing all of its leaves forms a path called its spine. The code of the caterpillar having spine 𝑃 𝑘 : 𝑣 1 𝑣 2 . . . 𝑣 𝑘 is the ordered 𝑘-tuple (ℓ1 , ℓ2 , . . . , ℓ𝑘 ), where ℓ𝑖 is the number of leaves in the caterpillar adjacent to 𝑣 𝑖 for 𝑖 ∈ [𝑘]. A rooted tree 𝑇 = 𝑇𝑟 distinguishes one vertex 𝑟 called the root and the tree 𝑇𝑟 is said to be rooted at vertex 𝑟. For each vertex 𝑣 ≠ 𝑟 of 𝑇𝑟 , the parent of 𝑣 is the neighbor of 𝑣 on the unique (𝑟, 𝑣)-path, while a child of 𝑣 is any other neighbor of 𝑣. © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_5

99

Chapter 5. Domination in Trees

100

The root 𝑟 does not have a parent in 𝑇𝑟 and all of its neighbors are its children. A descendant of 𝑣 is a vertex 𝑥 such that the unique (𝑟, 𝑥)-path contains 𝑣. Thus, every child of 𝑣 is a descendant of 𝑣. Let 𝐶 (𝑣) and 𝐷 (𝑣) denote the set of children and descendants, respectively, of 𝑣, and let 𝐷 [𝑣] = 𝐷 (𝑣) ∪ {𝑣}. The maximal subtree 𝑇𝑣 rooted at 𝑣 is the subtree induced by 𝐷 [𝑣].

5.2 Domination in Trees We begin with a simple observation about dominating sets in trees. If a 𝛾-set 𝑆 in a tree 𝑇 of order 𝑛 ≥ 3 contains a leaf, then we can simply replace this leaf in 𝑆 with its support vertex to produce another 𝛾-set of 𝑇. Thus, we have the following well-known result. Observation 5.1 If 𝑇 is a tree of order 𝑛 ≥ 3 with ℓ leaves, then the following properties hold: (a) There is a 𝛾-set of 𝑇 that contains no leaf. (b) 𝛾(𝑇) ≤ 𝑛 − ℓ. As observed in Chapter 2 and proven in Chapter 15, the three upper parameters of the Domination Chain are equal for bipartite graphs. Hence, the Domination Chain for domination and independence numbers of trees can be stated as follows. Theorem 5.2 If 𝑇 is a tree, then 𝛾(𝑇) ≤ 𝑖(𝑇) ≤ 𝛼(𝑇) = Γ(𝑇). For a simple example, the double star 𝑇 = 𝑆(𝑟, 𝑠), where 1 ≤ 𝑟 ≤ 𝑠, has 𝛾(𝑇) = 2, 𝑖(𝑇) = 𝑟 + 1, and 𝛼(𝑇) = Γ(𝑇) = 𝑟 + 𝑠. For instance, the double star 𝑇 = 𝑆(3, 4) shown in Figure 5.1 has 𝛾(𝑇) = 2, 𝑖(𝑇) = 4, and 𝛼(𝑇) = Γ(𝑇) = 7.

Figure 5.1 A double star 𝑆(3, 4)

Much research has focused on when equality is achieved between two parameters in the Domination Chain. As we shall see later in this chapter, the trees for which 𝛾(𝑇) = 𝑖(𝑇) have been characterized.

5.2.1

Domination Bounds in Trees

Recall that in Section 4.3.1, we presented Ore’s Theorem [622] from 1962. We also presented Theorem 4.24 from 1982 by Payan and Xuong [633] providing a characterization of the graphs achieving equality in Ore’s Theorem. Here we restate the characterization in the special case of trees.

Section 5.2. Domination in Trees

101

Theorem 5.3 ([622, 633]) If 𝑇 is tree of order 𝑛 ≥ 2, then 𝛾(𝑇) ≤ 21 𝑛, with equality if and only if 𝑇 = 𝑇 ′ ◦ 𝐾1 is the corona of some tree 𝑇 ′ . We note that Hansberg and Volkmann [377] characterized the trees of even order having domination number one less than half their order. Theorem 4.3 states Berge’s simple and tight upper bound on the domination number that for any graph 𝐺 of order 𝑛, 𝛾(𝐺) ≤ 𝑛 − Δ(𝐺). It is easy to see that this upper bound also holds for 𝑖(𝐺). In 1997 Domke et al. [235] characterized the connected bipartite graphs 𝐺 achieving equality in this bound for the domination parameters 𝛾(𝐺) and 𝑖(𝐺). Later in 1997, Favaron and Mynhardt [283] extended this work. Next we state the characterization from [235] of the trees attaining the bound for 𝛾(𝐺). A tree is a wounded spider if it can be obtained from a nontrivial star 𝐾1,𝑘 , where 𝑘 ≥ 1, by subdividing at most 𝑘 − 1 of its edges exactly once. Thus, a nontrivial star is a wounded spider. As a simple example, the wounded spider obtained from 𝐾1,6 by subdividing three of its edges is shown Figure 5.2. Let 𝑇 be a subdivided star obtained from a nontrivial star 𝐾1,𝑘 with center 𝑥 by subdividing 𝑗 edges, where 𝑗 ∈ [𝑘 − 1] 0 . Then 𝑥 along with the 𝑗 vertices of degree 2 created by the subdivision form a 𝛾-set (𝛾t -set) of 𝑇. Furthermore, the set consisting of 𝑥 together with the leaves adjacent to the vertices of degree 2 is an 𝑖-set. Thus, for this tree and all wounded spiders, 𝛾(𝑇) = 𝑖(𝑇) = 𝛾t (𝑇).

Figure 5.2 A wounded spider

Theorem 5.4 ([235]) If 𝑇 is tree of order 𝑛 ≥ 2, then 𝛾(𝑇) ≤ 𝑛 − Δ(𝑇), with equality if and only if 𝑇 is a wounded spider. In 2006 Blidia et al. [81] generalized Berge’s bound given in Theorem 4.3 as follows. For a vertex 𝑣 of 𝐺, let 𝛼𝑣′ (𝐺) denote the size of a largest matching in the graph 𝐺 − N[𝑣], that is, 𝛼𝑣′ (𝐺) = 𝛼′ (𝐺 − N[𝑣]). Let Δ′ (𝐺) denote the maximum of deg(𝑣) + 𝛼𝑣′ (𝐺) among all vertices 𝑣 of 𝐺. Theorem 5.5 ([81]) If 𝐺 is a graph, then 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 𝑛 − Δ′ (𝐺). The authors of [81] characterized the trees 𝑇 for which 𝛾(𝑇) = 𝑛 − Δ′ (𝑇) and those for which 𝑖(𝑇) = 𝑛 − Δ′ (𝑇). We refer the reader to their paper for these characterizations, but we will give examples of trees in these families here. For an example of a tree 𝑇 with 𝛾(𝑇) = 𝑛 − Δ′ (𝑇), see Figure 5.3 where the highlighted

Chapter 5. Domination in Trees

102 𝑣

Figure 5.3 A tree 𝑇 satisfying 𝛾(𝑇) = 11 = 𝑛 − Δ′ (𝑇)

vertices form a 𝛾-set of 𝑇. We note that 𝑇 has order 𝑛 = 26, 𝛾(𝑇) = 𝑖(𝑇) = 11, and Δ′ (𝑇) = 15, where for example, deg(𝑣) = 7 and 𝛼𝑣′ (𝑇) = 8. For an example of a tree 𝑇 ★ with 𝛾(𝑇 ★) < 𝑖(𝑇 ★) = 𝑛 − Δ′ (𝑇 ★), see Figure 5.4 where the highlighted vertices form an 𝑖-set of 𝑇 ★. We note that 𝑇 ★ has order 𝑛 = 27, 𝛼𝑣′ (𝑇 ★) = 8, 𝛾(𝑇 ★) = 11 < 12 = 𝑖(𝑇 ★), and Δ′ (𝑇 ★) = 15. 𝑣

Figure 5.4 A tree 𝑇 ★ with 𝑖(𝑇 ★) = 12 = 𝑛 − Δ′ (𝑇 ★) Next we present a lower bound on the domination number of a tree. Vertices of maximum eccentricity are called peripheral. Let ecc★ (𝐺) denote the maximum distance from the set of peripheral vertices of 𝐺 to a vertex not in the set, where the distance from a vertex to a set is the smallest distance from the vertex to any of the vertices in the set. DeLaViña et al. [222] proved a lower bound on the domination number of a tree 𝑇 in terms ecc★ (𝑇).
Theorem 5.6 ([222]) If 𝑇 is a nontrivial tree, then 𝛾(𝑇) ≥ 12 1 + ecc★ (𝑇) . Theorem 5.6 is tight as can be seen with paths of odd order. Although they only proved it for trees, the authors of [222] believe that the bound of Theorem 5.6 holds for general graphs. Let 𝛼L′ (𝐺) denote the minimum cardinality of a maximal matching of 𝐺, also called the lower matching number in the literature. We note that 𝛼L′ (𝐺) is not as well-studied as the cardinality of a maximum matching 𝛼′ (𝐺). DeLaViña et al. [222] also determined upper and lower bounds on the domination number of a tree 𝑇 in terms of 𝛼L′ (𝑇). Theorem 5.7 ([222]) If 𝑇 is a tree, then 𝛼L′ (𝑇) ≤ 𝛾(𝑇) ≤ 2𝛼L′ (𝑇).

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103

The bounds stated in Theorem 5.7 are tight. For example, if 𝑇 is the path 𝑃𝑛 for 𝑛 ≡ 0 (mod 3), then 𝛾(𝑇) = 𝛼L′ (𝑇) = 𝑛3 . For the upper bound, if 𝑇 is the corona of an even path 𝑃2𝑘 for 𝑘 ≥ 1, then 𝛾(𝑇) = 2𝑘 = 2𝛼L′ (𝑇). We note that the lower bound of Theorem 5.7 is not true for graphs in general. In fact, the difference 𝛼L′ (𝐺) − 𝛾(𝐺) can be made arbitrarily large. Consider the graph 𝐺 𝑚 formed from 𝑚𝐾2 labeled 𝑢 𝑖 𝑣 𝑖 for 𝑖 ∈ [𝑚] by adding two new vertices 𝑢 and 𝑣 and edges 𝑢𝑢 𝑖 , 𝑣𝑣 𝑖 for 𝑖 ∈ [𝑚]. The resulting graph 𝐺 𝑚 satisfies 𝛾(𝐺 𝑚 ) = 2, while 𝛼L′ (𝐺 𝑚 ) = 𝑚.

5.2.2 Domination Lower Bounds Involving the Number of Leaves Let R be the family of trees in which the distance between any two distinct leaves is congruent to 2 modulo 3. As an illustration, the caterpillar 𝑇 with code (2, 0, 0, 3, 0, 0, 3, 0, 0, 2) shown in Figure 5.5 is an example of a tree that belongs to the family R. In this example, 𝑇 has order 𝑛 = 20, ℓ = 10 leaves, and 𝛾(𝑇) = 4 = 13 (𝑛 − ℓ + 2), where the four highlighted vertices in Figure 5.5 form a 𝛾-set of 𝑇.

Figure 5.5 A tree 𝑇 in the family R

Lemańska [557] established the following lower bound on the domination number of a tree in terms of its order and number of leaves, and characterized the family of trees achieving this bound. Theorem 5.8 ([557]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then 𝛾(𝑇) ≥ 1 3 (𝑛 − ℓ + 2), with equality if and only if 𝑇 ∈ R. Proof Let 𝑇 be a tree of order 𝑛 ≥ 3. We proceed by induction on 𝑛. If 𝑛 = 3, then 𝑇 is the path 𝑃3 ∈ R and the result is immediate, establishing the base case. Let 𝑇 be a tree of order 𝑛 ≥ 4 and assume that if 𝑇 ′ is a tree with ℓ ′ leaves and order 𝑛′ , where 3 ≤ 𝑛′ < 𝑛, then 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 2), with equality if and only if 𝑇 ′ ∈ R. By Observation 5.1(a), there is a 𝛾-set 𝑆 of 𝑇 that contains no leaf of 𝑇. Among all paths in 𝑇 with length diam(𝑇) = 𝑑, let 𝑣 0 𝑣 1 . . . 𝑣 𝑑 be one such that deg𝑇 (𝑣 1 ) is maximized. Root the tree 𝑇 at the vertex 𝑟 = 𝑣 𝑑 . Suppose that deg𝑇 (𝑣 1 ) > 2. In this case, we let 𝑇 ′ = 𝑇 − 𝑣 0 with order 𝑛′ = 𝑛 − 1. Since 𝑣 1 is a support vertex in 𝑇 ′ , it follows that ℓ ′ = ℓ − 1 and 𝛾(𝑇 ′ ) = 𝛾(𝑇). By our inductive hypothesis, 𝛾(𝑇) = 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 2) = 13 (𝑛 − ℓ + 2).

(5.1)

Moreover, if 𝛾(𝑇) = 13 (𝑛 − ℓ + 2), then we have equality throughout Inequality (5.1) and so 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 2), implying that 𝑇 ′ ∈ R. Let 𝑢 0 be a child of 𝑣 1 different from 𝑣 0 . Now 𝑑𝑇 (𝑢 0 , 𝑣 0 ) = 2. Since 𝑢 0 and 𝑣 0 are at the same distance from every

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other leaf in 𝑇, and 𝑇 ′ ∈ R, the tree 𝑇 therefore also belongs to R. Hence, we may assume that deg𝑇 (𝑣 1 ) = 2, for otherwise the desired result follows. Suppose that deg𝑇 (𝑣 2 ) > 2. In this case, we let 𝑇 ′ = 𝑇 − {𝑣 0 , 𝑣 1 } with order ′ 𝑛 = 𝑛−2. We note that ℓ ′ = ℓ−1. Let 𝑢 1 be a child of 𝑣 2 different from 𝑣 1 . If 𝑢 1 is a leaf, then 𝑣 2 ∈ 𝑆. If 𝑢 1 is not a leaf, then it is a support vertex of degree 2 and 𝑢 1 ∈ 𝑆. In both cases, the set 𝑆\{𝑣 1 } is a dominating set of 𝑇 ′ , and so 𝛾(𝑇 ′ ) ≤ |𝑆|−1 = 𝛾(𝑇)−1. By our inductive hypothesis, 𝛾(𝑇) ≥ 𝛾(𝑇 ′ )+1 ≥ 31 (𝑛′ −ℓ ′ +2)+1 = 13 (𝑛−ℓ+4) > 13 (𝑛−ℓ+2). Hence, we may assume that deg𝑇 (𝑣 2 ) = 2, for otherwise the lower bound holds with strict inequality. With our earlier assumptions, if 𝑛 ∈ {4, 5}, then 𝑇 is a path 𝑃𝑛 , and 𝛾(𝑇) = 2 > 13 𝑛 = 13 (𝑛 − ℓ + 2). Hence, we may assume that 𝑛 ≥ 6. Therefore, let 𝑇 ′ = 𝑇 − {𝑣 0 , 𝑣 1 , 𝑣 2 } with order 𝑛′ = 𝑛 − 3 ≥ 3. If deg𝑇 (𝑣 3 ) > 2, then ℓ ′ = ℓ − 1. If deg𝑇 (𝑣 3 ) = 2, then 𝑣 3 is a leaf in 𝑇 ′ and ℓ ′ = ℓ. In both cases, ℓ ′ ≤ ℓ. Recall that 𝑣 1 ∈ 𝑆. If 𝑣 2 ∈ 𝑆, then we can replace 𝑣 2 in 𝑆 with the vertex 𝑣 3 . Hence, we may assume that 𝑣 2 ∉ 𝑆, implying that the set 𝑆 \ {𝑣 1 } is a dominating set of 𝑇 ′ , whence 𝛾(𝑇 ′ ) ≤ |𝑆| −1 = 𝛾(𝑇) −1. By our inductive hypothesis, 𝛾(𝑇) ≥ 𝛾(𝑇 ′ ) + 1 ≥ 13 (𝑛′ − ℓ ′ + 2) + 1 ≥ 13 (𝑛 − ℓ + 2), noting that 𝑛′ = 𝑛 − 3 and ℓ ′ ≤ ℓ. Moreover, if 𝛾(𝑇) = 13 (𝑛 − ℓ + 2), then we have equality throughout the previous inequality chain, implying that 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 2) and ℓ ′ = ℓ. This in turn implies that 𝑇 ′ ∈ R and the vertex 𝑣 3 is a leaf in 𝑇 ′ . Since 𝑇 ′ ∈ R, the distance between any two distinct leaves in 𝑇 ′ is congruent to 2 modulo 3. Let 𝑢 and 𝑣 be two leaves in 𝑇. If both 𝑢 and 𝑣 belong to 𝑇 ′ , then 𝑑𝑇 (𝑢, 𝑣) = 𝑑𝑇 ′ (𝑢, 𝑣) ≡ 2 (mod 3). If one of 𝑢 and 𝑣 does not belong to 𝑇 ′ , then we may assume that 𝑢 = 𝑣 0 . In this case, 𝑑𝑇 (𝑢, 𝑣) = 𝑑𝑇 (𝑢, 𝑣 3 ) + 𝑑𝑇 (𝑣 3 , 𝑣) = 3 + 𝑑𝑇 ′ (𝑣 3 , 𝑣) ≡ 2 (mod 3) noting that the distance between 𝑣 3 and 𝑣 in 𝑇 ′ is congruent to 2 modulo 3. Thus, the distance between any two distinct leaves in 𝑇 is congruent to 2 modulo 3, that is, 𝑇 ∈ R. In 2019 Hajian et al. [372] defined a family G00 of trees as follows. Definition 5.9 Let G00 be the family of trees 𝑇 that can be obtained from a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 of trees, where 𝑘 ≥ 1, such that 𝑇1 is a star with at least three vertices, 𝑇 = 𝑇𝑘 , and, if 𝑘 ≥ 2, then the tree 𝑇𝑖+1 can be obtained from the tree 𝑇𝑖 by applying Operation O defined below, for all 𝑖 ∈ [𝑘 − 1]. Operation O. Add a star 𝑄 𝑖 with at least three vertices to the tree 𝑇𝑖 and add an edge joining a leaf of 𝑄 𝑖 and a leaf of 𝑇𝑖 . Hajian et al. [372] showed that G00 ⊆ R and R ⊆ G00 , that is, the family R is precisely the family G00 . Theorem 5.8 due to Lemańska [557] can therefore be restated as follows. Theorem 5.10 ([557]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then 𝛾(𝑇) ≥ 1 0 3 (𝑛 − ℓ + 2), with equality if and only if 𝑇 ∈ G0 . We shall need the following elementary property of graphs in the family G00 . Proposition 5.11 If 𝑇 ∈ G00 is obtained from a star with at least three vertices by applying Operation O exactly 𝑘 − 1 times for some 𝑘 ≥ 1, then 𝛾(𝑇) = 𝜌(𝑇) = 𝑘. Further, 𝛾(𝑇 − 𝑣) ≥ 𝛾(𝑇) for every vertex 𝑣 in 𝑇.

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In order to generalize the result of Lemańska [557], the authors in [372] defined two additional families G01 and G02 of trees, as follows. Definition 5.12 Let T01,1 be the family of trees 𝑇 that can be obtained from a tree 𝑇 ′ ∈ G00 by adding a star with at least three vertices and adding an edge from a leaf of the added star to a non-leaf in 𝑇 ′ . Let G01 be the family of trees 𝑇 that can be obtained from a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 of trees where 𝑘 ≥ 1 such that 𝑇1 ∈ T01,1 ∪ {𝑃2 }, 𝑇 = 𝑇𝑘 , and, if 𝑘 ≥ 2, then the tree 𝑇𝑖+1 can be obtained from the tree 𝑇𝑖 by applying Operation O (see Definition 5.9) for all 𝑖 ∈ [𝑘 − 1]. The following lemma determines the domination number of a graph that belongs to the family G01 . The proofs we present of Lemma 5.13 and Theorem 5.14 are from [372]. Lemma 5.13 ([372]) If 𝑇 ∈ G01 has order 𝑛 ≥ 2 and ℓ leaves, then 𝛾(𝑇) = 1 3 (𝑛 − ℓ + 3). Proof Let 𝑇 ∈ G01 be a tree of order 𝑛 ≥ 2 with ℓ leaves. The tree 𝑇 can be obtained from a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 of trees where 𝑘 ≥ 1 such that 𝑇1 ∈ T01,1 ∪ {𝑃2 }, 𝑇 = 𝑇𝑘 , and, if 𝑘 ≥ 2, then the tree 𝑇𝑖+1 can be obtained from the tree 𝑇𝑖 by applying Operation O for all 𝑖 ∈ [𝑘 − 1]. We prove by induction on 𝑘 ≥ 1 that 𝛾(𝑇) = 13 (𝑛 − ℓ + 3). Suppose that 𝑘 = 1. Thus, either 𝑇 = 𝑃2 or 𝑇 ∈ T01,1 . If 𝑇 = 𝑃2 , then 𝑛 = ℓ = 2 and 𝛾(𝑇) = 1 = 13 (𝑛 − ℓ + 3). Hence, we may assume that 𝑇 ∈ T01,1 . Thus, 𝑇 can be obtained from a tree 𝑇 ′ ∈ G00 by adding a star 𝑄 with 𝑡 ≥ 3 vertices and adding an edge from a leaf of the added star to a non-leaf in 𝑇 ′ . We note that 𝛾(𝑇) = 𝛾(𝑇 ′ ) + 1. Let 𝑇 ′ have order 𝑛′ and ℓ ′ leaves. By construction of the tree 𝑇, we note that 𝑛 = 𝑛′ + 𝑡 and ℓ = ℓ ′ + 𝑡 − 2. By Theorem 5.10, 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 2) = 13 (𝑛 − ℓ). Thus, 𝛾(𝑇) = 𝛾(𝑇 ′ ) + 1 = 13 (𝑛 − ℓ + 3). This establishes the base case. Suppose that 𝑘 ≥ 2 and let 𝑇 ★ = 𝑇𝑘−1 . Let 𝑇 ★ have order 𝑛★ and ℓ★ leaves. Applying the induction hypothesis to the tree 𝑇 ★ ∈ G01 , we have 𝛾(𝑇 ★) = 13 (𝑛★ −ℓ★ +3). The tree 𝑇 can be obtained from 𝑇 ★ by adding a star 𝑄★ with 𝑞 ≥ 3 vertices and adding an edge from a leaf of 𝑄★ to a leaf of 𝑇 ★. By construction of the tree 𝑇, we note that 𝛾(𝑇) = 𝛾(𝑇 ★) + 1. Further, 𝑛 = 𝑛★ + 𝑞 and ℓ = (ℓ★ − 1) + (𝑞 − 2). Thus, 𝛾(𝑇) = 𝛾(𝑇 ★) + 1 = 13 (𝑛★ − ℓ★ + 3) + 1 = 13 (𝑛 − ℓ) + 1 = 13 (𝑛 − ℓ + 3). Theorem 5.14 ([372]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then 𝛾(𝑇) = 1 1 3 (𝑛 − ℓ + 3) if and only if 𝑇 ∈ G0 . Proof The sufficiency follows from Lemma 5.13. To prove the necessity, let 𝑇 be a tree of order 𝑛 ≥ 2 with ℓ leaves satisfying 𝛾(𝑇) = 13 (𝑛 − ℓ + 3). We proceed by induction on 𝑛 ≥ 2 to show that 𝑇 ∈ G01 . If 𝑛 = 2, then 𝑇 = 𝑃2 ∈ G01 . This establishes the base case. Let 𝑛 ≥ 3 and assume that if 𝑇 ′ is a tree of order 𝑛′ ≥ 2, where 𝑛′ < 𝑛 with ℓ ′ leaves such that 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 3), then 𝑇 ′ ∈ G01 . Let 𝑇 be a tree of order 𝑛 with ℓ leaves satisfying 𝛾(𝑇) = 13 (𝑛 − ℓ + 3). By Observation 5.1(a), there is a 𝛾-set 𝑆 of 𝑇 that contains no leaf of 𝑇.

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If 𝑇 is a star, then 𝑇 ∈ G00 and by Theorem 5.10, 𝛾(𝑇) = 31 (𝑛 − ℓ + 2), a contradiction. If 𝑇 is a double star, then ℓ = 𝑛 − 2 and 𝛾(𝑇) = 2 > 13 (𝑛 − ℓ + 3), a contradiction. Hence, diam(𝑇) ≥ 4 and 𝑛 ≥ 5. Let 𝑣 0 𝑣 1 . . . 𝑣 𝑑 be a longest path in 𝑇. Root the tree 𝑇 at the vertex 𝑟 = 𝑣 𝑑 . Since the 𝛾-set 𝑆 contains no leaf of 𝑇, we note that 𝑣 1 ∈ 𝑆 and no child of 𝑣 1 belongs to 𝑆. Let deg𝑇 (𝑣 1 ) = 𝑡. Since {𝑣 0 , 𝑣 2 } ⊆ N𝑇 (𝑣 1 ), we note that 𝑡 ≥ 2. We show that deg𝑇 (𝑣 2 ) = 2. Suppose, to the contrary, that deg𝑇 (𝑣 2 ) ≥ 3. We now consider the tree 𝑇 ′ obtained from 𝑇 by removing the maximal subtree at 𝑇𝑣1 , that is, 𝑇 ′ = 𝑇 − 𝑉 (𝑇𝑣1 ). Let 𝑇 ′ have order 𝑛′ with ℓ ′ leaves. By construction of the tree 𝑇, we note that 𝑛 = 𝑛′ + 𝑡 and ℓ = ℓ ′ + 𝑡 − 1. By Theorem 5.10, 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 2) = 13 (𝑛 − ℓ + 1). Let 𝑢 1 be a child of 𝑣 2 in 𝑇 different from 𝑣 1 . If 𝑢 1 is a leaf, then by our choice of the set 𝑆, we note that 𝑣 2 ∈ 𝑆. If 𝑢 1 is not a leaf, then every child of 𝑢 1 is a leaf, and so 𝑢 1 ∈ 𝑆. In both cases, the set 𝑆 \ {𝑣 1 } is a dominating set of 𝑇 ′ , implying that 𝛾(𝑇 ′ ) ≤ |𝑆| − 1 = 13 (𝑛 − ℓ + 3) − 1 = 13 (𝑛 − ℓ), contradicting our earlier observation that 𝛾(𝑇 ′ ) ≥ 13 (𝑛 − ℓ + 1). Hence, deg𝑇 (𝑣 2 ) = 2. Suppose next that deg𝑇 (𝑣 3 ) ≥ 3. We now consider the tree 𝑇 ′ obtained from 𝑇 by removing the maximal subtree at 𝑇𝑣2 , that is, 𝑇 ′ = 𝑇 − 𝑉 (𝑇𝑣2 ). Let 𝑇 ′ have order 𝑛′ with ℓ ′ leaves. Recall that 𝑣 1 ∈ 𝑆. If 𝑣 2 ∈ 𝑆, then we can simply replace 𝑣 2 in 𝑆 by its parent 𝑣 3 . Hence, we may assume that 𝑣 1 is the only vertex of 𝑆 that belongs to 𝑉 (𝑇𝑣2 ). This implies that the set 𝑆 \ {𝑣 1 } is a dominating set of 𝑇 ′ , and so 𝛾(𝑇 ′ ) ≤ |𝑆| − 1 = 13 (𝑛 − ℓ + 3) − 1 = 13 (𝑛 − ℓ). By construction of the tree 𝑇, we note that 𝑛 = 𝑛′ + 𝑡 + 1 and ℓ = ℓ ′ + 𝑡 − 1. By Theorem 5.10, we have 1 1 ′ 1 ′ 1 ′ ′ ′ ′ 3 (𝑛 − ℓ) ≥ 𝛾(𝑇 ) ≥ 3 (𝑛 − ℓ + 2) = 3 (𝑛 − ℓ), implying that 𝛾(𝑇 ) = 3 (𝑛 − ℓ + 2). ′ 0 ′ By Theorem 5.10, the tree 𝑇 ∈ G0 . Thus, the tree 𝑇 is obtained from 𝑇 ∈ G00 by attaching the star 𝑇𝑣2 (centered at 𝑣 1 ) and adding an edge from the leaf 𝑣 2 of the added star 𝑇𝑣2 to the non-leaf 𝑣 3 of 𝑇 ′ , implying that 𝑇 ∈ T01,1 ⊂ G01 . We may therefore assume that deg𝑇 (𝑣 3 ) = 2, for otherwise 𝑇 ∈ G01 as desired. We now consider the tree 𝑇 ′ = 𝑇 − 𝑉 (𝑇𝑣2 ). Every 𝛾-set of 𝑇 ′ can be extended to a dominating set of 𝑇 by adding to it vertex 𝑣 1 , and so 𝛾(𝑇) ≤ 𝛾(𝑇 ′ ) + 1. Let 𝑇 ′ have order 𝑛′ with ℓ ′ leaves. By construction of the tree 𝑇, we note that 𝑛 = 𝑛′ + 𝑡 + 1 and ℓ = (ℓ ′ −1) + (𝑡 −1) = ℓ ′ +𝑡 −2. By Theorem 5.10, 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ −ℓ ′ +2) = 13 (𝑛−ℓ−1). If 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 2), then 𝛾(𝑇) ≤ 𝛾(𝑇 ′ ) + 1 = 13 (𝑛 − ℓ − 1) + 1 = 13 (𝑛 − ℓ + 2), a contradiction. Thus, 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 3) = 13 (𝑛 − ℓ). Hence, 13 (𝑛 − ℓ + 3) = 𝛾(𝑇) = 𝛾(𝑇 ′ ) + 1 ≥ 31 (𝑛 − ℓ + 3), implying that 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 3). Applying the inductive hypothesis to the tree 𝑇 ′ , we have 𝑇 ′ ∈ G01 . Thus, the tree 𝑇 is obtained from the tree 𝑇 ′ ∈ G01 by applying Operation O, implying that 𝑇 ∈ G01 . We consider next the family G02 defined in [372]. Definition 5.15 Let T02,1 be the family of trees 𝑇 that can be obtained from a tree 𝑇 ′ ∈ G00 by adding a star (with at least two vertices) and adding an edge from the center of the added star to a non-leaf in 𝑇 ′ . Let T02,2 be the family of trees 𝑇 that can be obtained from a tree 𝑇 ′ ∈ G01 by adding a star with at least three vertices and adding an edge from a leaf of the added star to a non-leaf in 𝑇 ′ . Now, let G02 be the

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107

family of trees 𝑇 that can be obtained from a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 of trees, where 𝑘 ≥ 1, such that 𝑇1 ∈ T02,1 ∪ T02,2 ∪ {𝑃4 }, 𝑇 = 𝑇𝑘 , and if 𝑘 ≥ 2, then the tree 𝑇𝑖+1 can be obtained from the tree 𝑇𝑖 by applying Operation O (see Definition 5.9) for all 𝑖 ∈ [𝑘 − 1]. The following lemma determines the domination number of a graph that belongs to the family G02 . The proofs we present of Lemma 5.16 and Theorem 5.17 are from [372]. Lemma 5.16 ([372]) If 𝑇 ∈ G02 has order 𝑛 ≥ 2 and ℓ leaves, then 𝛾(𝑇) = 1 3 (𝑛 − ℓ + 4). Proof Let 𝑇 ∈ G02 be a tree of order 𝑛 ≥ 2 with ℓ leaves. The tree 𝑇 can be obtained from a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 of trees, where 𝑘 ≥ 1 such that 𝑇1 ∈ T02,1 ∪T02,2 ∪{𝑃4 }, 𝑇 = 𝑇𝑘 , and, if 𝑘 ≥ 2, then the tree 𝑇𝑖+1 can be obtained from the tree 𝑇𝑖 by applying Operation O for all 𝑖 ∈ [𝑘 −1]. We prove by induction on 𝑘 ≥ 1 that 𝛾(𝑇) = 13 (𝑛−ℓ+4). If 𝑘 = 1, then either 𝑇 ∈ T02,1 ∪ T02,2 ∪ {𝑃4 }. If 𝑇 = 𝑃4 , then 𝑛 = 4, ℓ = 2, and 𝛾(𝑇) = 2 = 13 (𝑛 − ℓ + 4). Hence, we may assume that 𝑇 ∈ T02,1 ∪ T02,2 . Assume that 𝑇 ∈ T02,1 . In this case, 𝑇 can be obtained from a tree 𝑇 ′ ∈ G00 by adding a star with 𝑡 ≥ 3 vertices and adding an edge from the center of the added star to a non-leaf in 𝑇 ′ . By Proposition 5.11, we have 𝛾(𝑇) = 𝛾(𝑇 ′ ) + 1. Let 𝑇 ′ have order 𝑛′ and ℓ ′ leaves. By construction of the tree 𝑇, 𝑛 = 𝑛′ + 𝑡 and ℓ = ℓ ′ + 𝑡 − 1. By Theorem 5.10, 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 2) = 13 (𝑛 − ℓ + 1). Thus, 𝛾(𝑇) = 𝛾(𝑇 ′ ) + 1 = 13 (𝑛 − ℓ + 4). Assume that 𝑇 ∈ T02,2 . In this case, 𝑇 can be obtained from a tree 𝑇 ′ ∈ G01 by adding a star with 𝑡 ≥ 3 vertices and adding an edge from a leaf of the added star to a non-leaf in 𝑇 ′ . We note that 𝛾(𝑇) = 𝛾(𝑇 ′ ) + 1. Let 𝑇 ′ have order 𝑛′ with ℓ ′ leaves. By construction of the tree 𝑇, we have 𝑛 = 𝑛′ + 𝑡 and ℓ = ℓ ′ + 𝑡 − 2. By Lemma 5.13, 𝛾(𝑇 ′ ) = 13 (𝑛′ − ℓ ′ + 3) = 13 (𝑛 − ℓ + 1). Thus, 𝛾(𝑇) = 𝛾(𝑇 ′ ) + 1 = 13 (𝑛 − ℓ + 4). This establishes the base case. Suppose that 𝑘 ≥ 2 and let 𝑇 ★ = 𝑇𝑘−1 . Let 𝑇 ★ have order 𝑛★ and ℓ★ leaves. Applying the induction hypothesis to the tree 𝑇 ★ ∈ G02 , we have 𝛾(𝑇 ★) = 13 (𝑛★ −ℓ★ +4). The tree 𝑇 can be obtained from 𝑇 ★ by adding a star with 𝑞 ≥ 3 vertices and adding an edge from a leaf of the added star to a leaf of 𝑇 ★. By construction of the tree 𝑇, we have 𝛾(𝑇) = 𝛾(𝑇 ★) + 1. Further, 𝑛 = 𝑛★ + 𝑞 and ℓ = (ℓ★ − 1) + (𝑞 − 2). Thus, 𝛾(𝑇) = 𝛾(𝑇 ★) + 1 = 13 (𝑛★ − ℓ★ + 4) + 1 = 13 (𝑛 − ℓ + 1) + 1 = 13 (𝑛 − ℓ + 4). Theorem 5.17 ([372]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then 𝛾(𝑇) = 1 2 3 (𝑛 − ℓ + 4) if and only if 𝑇 ∈ G0 . Proof The sufficiency follows from Lemma 5.16. To prove the necessity, let 𝑇 be a tree of order 𝑛 ≥ 2 with ℓ leaves satisfying 𝛾(𝑇) = 13 (𝑛 − ℓ + 4). We proceed by induction on 𝑛 ≥ 2 to show that 𝑇 ∈ G02 . If 𝑛 = 2, then 𝑇 = 𝑃2 and 𝛾(𝑇) = 1 = 13 (𝑛 − ℓ + 3), a contradiction. If 𝑇 is a star, then 𝑇 ∈ G00 and by Theorem 5.10, 𝛾(𝑇) = 13 (𝑛 − ℓ + 2), a contradiction. If 𝑇 = 𝑃4 , then 𝑇 ∈ G02 . This establishes the base case. Let 𝑛 ≥ 5 and assume that if 𝑇 ′ is a tree of order 𝑛′ ≥ 2, where 𝑛′ < 𝑛

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108

with ℓ ′ leaves such that 𝛾(𝑇 ′ ) = 31 (𝑛′ − ℓ ′ + 4), then 𝑇 ′ ∈ G02 . Let 𝑇 be a tree of order 𝑛 with ℓ leaves satisfying 𝛾(𝑇) = 13 (𝑛 − ℓ + 4). By Observation 5.1(a), there is a 𝛾-set 𝑆 of 𝑇 that contains no leaf of 𝑇. If 𝑇 is a double star, then 𝑇 ∈ T02,1 ⊆ G02 . Hence, we may assume that diam(𝑇) ≥ 4. Let 𝑣 0 𝑣 1 . . . 𝑣 𝑑 be a longest path in 𝑇. We now root the tree 𝑇 at the vertex 𝑟 = 𝑣 𝑑 . Since the 𝛾-set 𝑆 contains no leaf of 𝑇, we note that 𝑣 1 ∈ 𝑆 and no child of 𝑣 1 belongs to 𝑆. Let deg𝑇 (𝑣 1 ) = 𝑡, where 𝑡 ≥ 2. Suppose that deg𝑇 (𝑣 2 ) ≥ 3. We now consider the tree 𝑇 ′ = 𝑇 − 𝑉 (𝑇𝑣1 ). Let 𝑇 ′ have order 𝑛′ with ℓ ′ leaves. Let 𝑢 1 be a child of 𝑣 2 different from 𝑣 1 . If 𝑢 1 is a leaf, then 𝑣 2 ∈ 𝑆. If 𝑢 1 is not a leaf, then 𝑢 1 is a support vertex and 𝑢 1 ∈ 𝑆. In both cases, the vertex 𝑣 2 is dominated by 𝑆 \ {𝑣 1 }, implying that 𝑆 \ {𝑣 1 } is a dominating set of 𝑇 ′ , and so 𝛾(𝑇 ′ ) ≤ |𝑆| − 1 = 13 (𝑛 − ℓ + 4) − 1 = 13 (𝑛 − ℓ + 1). By construction of the tree 𝑇, we have 𝑛 = 𝑛′ + 𝑡 and ℓ = ℓ ′ + 𝑡 − 1. By Theorem 5.10, 1 3 (𝑛

− ℓ + 1) ≥ 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 2) = 13 (𝑛 − ℓ + 1).

(5.2)

Since we must have equality throughout Inequality (5.2), we have 𝛾(𝑇 ′ ) = 13 (𝑛′ −ℓ ′ +2). By Theorem 5.10, the tree 𝑇 ′ is in G00 . Thus, the tree 𝑇 is obtained from the tree 𝑇 ′ ∈ G00 by attaching the star 𝑇𝑣1 centered at 𝑣 1 (possibly, 𝑇𝑣1 = 𝑃2 ) and adding an edge from the center 𝑣 1 of the added star 𝑇𝑣1 to the non-leaf 𝑣 2 of 𝑇 ′ , implying that 𝑇 ∈ T02,1 ⊂ G02 . We may assume therefore that deg𝑇 (𝑣 2 ) = 2, for otherwise 𝑇 ∈ G02 as desired. Suppose next that deg𝑇 (𝑣 3 ) ≥ 3, and consider the tree 𝑇 ′ = 𝑇 −𝑉 (𝑇𝑣2 ). Let 𝑇 ′ have order 𝑛′ with ℓ ′ leaves. If 𝑣 2 ∈ 𝑆, then we can simply replace 𝑣 2 in 𝑆 by its parent 𝑣 3 . Hence, we may assume that 𝑣 1 is the only vertex of 𝑆 that belongs to 𝑉 (𝑇𝑣2 ), implying that the set 𝑆\{𝑣 1 } is a dominating set of 𝑇 ′ , and so 𝛾(𝑇 ′ ) ≤ |𝑆| −1 = 13 (𝑛−ℓ+4) −1 = 1 ′ ′ 3 (𝑛 − ℓ + 1). By construction of the tree 𝑇, we have 𝑛 = 𝑛 + 𝑡 + 1 and ℓ = ℓ + 𝑡 − 1. If 1 ′ ′ ′ ′ 𝛾(𝑇 ) = 3 (𝑛 − ℓ + 2), then since every 𝛾-set of 𝑇 can be extended to a dominating set of 𝑇 by adding 𝑣 1 to it, 𝛾(𝑇) ≤ 𝛾(𝑇 ′ ) + 1 = 13 (𝑛′ − ℓ ′ + 2) + 1 = 13 (𝑛 − ℓ + 3), a contradiction. Hence, by Theorem 5.10, 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 3). Thus, by our earlier observations, 1 3 (𝑛

− ℓ + 1) ≥ 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 3) = 13 (𝑛 − ℓ + 1).

(5.3)

Since we must have equality throughout Inequality (5.3), we have 𝛾(𝑇 ′ ) = 13 (𝑛′ −ℓ ′ +3). By Theorem 5.14, the tree 𝑇 ′ is in G01 . Thus, the tree 𝑇 is obtained from the tree 𝑇 ′ ∈ G01 by attaching the star 𝑇𝑣2 , centered at 𝑣 1 , to 𝑇 ′ by adding an edge between the leaf 𝑣 2 of the star 𝑇𝑣2 to the non-leaf vertex 𝑣 3 of 𝑇 ′ , implying that 𝑇 ∈ T02,2 ⊂ G02 . We may assume therefore that deg𝑇 (𝑣 3 ) = 2, for otherwise 𝑇 ∈ G02 as desired. We again consider the tree 𝑇 ′ = 𝑇 − 𝑉 (𝑇𝑣2 ). As before, we may assume that 𝑣 1 is the only vertex of 𝑆 that belongs to 𝑉 (𝑇𝑣2 ), implying that 𝑆 \ {𝑣 1 } is a dominating set of 𝑇 ′ , and so 𝛾(𝑇 ′ ) ≤ |𝑆| − 1 = 13 (𝑛 − ℓ + 4) − 1 = 13 (𝑛 − ℓ + 1). Every 𝛾-set of 𝑇 ′ can be extended to a dominating set of 𝑇 by adding to it the vertex 𝑣 1 , and so 𝛾(𝑇) ≤ 𝛾(𝑇 ′ ) + 1. Let 𝑇 ′ have order 𝑛′ with ℓ ′ leaves. By construction of the tree 𝑇, we have 𝑛 = 𝑛′ + 𝑡 + 1 and ℓ = (ℓ ′ − 1) + (𝑡 − 1) = ℓ ′ + 𝑡 − 2. If 𝛾(𝑇 ′ ) ≤ 13 (𝑛′ − ℓ ′ + 3),

Section 5.2. Domination in Trees

109

then 𝛾(𝑇) ≤ 𝛾(𝑇 ′ ) + 1 ≤ 31 (𝑛′ − ℓ ′ + 3) + 1 = 13 (𝑛 − ℓ + 3), a contradiction. Hence, by Theorem 5.10, 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 4). Thus, by our earlier observations, 1 3 (𝑛

− ℓ + 1) ≥ 𝛾(𝑇 ′ ) ≥ 13 (𝑛′ − ℓ ′ + 4) = 13 (𝑛 − ℓ + 1).

(5.4)

Since we must have equality throughout Inequality (5.4), we have 𝛾(𝑇 ′ ) = 13 (𝑛′ −ℓ ′ +4). Applying the inductive hypothesis to the tree 𝑇 ′ , we have 𝑇 ′ is in G02 . Thus, the tree 𝑇 is obtained from the tree 𝑇 ′ ∈ G02 by applying Operation O, implying that 𝑇 ∈ G02 . The results of Theorems 5.10, 5.14, and 5.17 are summarized in the following theorem. Theorem 5.18 ([372]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then   𝑛−ℓ+2 𝛾(𝑇) ≥ , 3 with equality if and only if 𝑇 ∈ G00 ∪ G01 ∪ G02 . We remark that Theorem 5.18 was generalized in [372] to connected graphs of order 𝑛 ≥ 2 with 𝑘 ≥ 0 cycles and ℓ leaves. In order to state this stronger result, the authors in [372] defined the families G𝑘0 , G𝑘1 , and G𝑘2 of graphs for all integers 𝑘 ≥ 0, where in the special case when 𝑘 = 0, the families G00 , G01 , and G02 are defined earlier in Definitions 5.9, 5.12, and 5.15, respectively. However, we omit the definitions of the families G𝑘0 , G𝑘1 , and G𝑘2 for 𝑘 ≥ 1, since this chapter focuses only on domination in trees. We are now in a position to state the main result in [372] that generalizes Theorem 5.18. In the special case when 𝑘 = 0, Theorem 5.19 is a restatement of the result of Theorem 5.18. Theorem 5.19 ([372]) If 𝐺 is a connected graph of order 𝑛 ≥ 2 with 𝑘 ≥ 0 cycles and ℓ leaves, then   𝑛 − ℓ + 2(1 − 𝑘) 𝛾(𝐺) ≥ , 3 with equality if and only if 𝐺 ∈ G𝑘0 ∪ G𝑘1 ∪ G𝑘2 .

5.2.3

Slater Lower Bound on the Domination Number

As we have seen in Theorem 4.9 in Chapter 4, Slater [680] determined a lower bound on the domination number of a graph 𝐺 in terms of a parameter, which is now known as the Slater number, based on the non-increasing degree sequence of 𝐺. Recall that the Slater number sl(𝐺) of a graph 𝐺 is the smallest integer 𝑡 such that 𝑡 added to the sum of the first 𝑡 terms of the non-increasing degree sequence of 𝐺 is at least as large as the number of vertices of 𝐺. We repeat Theorem 4.9. Theorem 5.20 ([680]) For any graph 𝐺, 𝛾(𝐺) ≥ sl(𝐺).

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Desormeaux et al. [227] in 2014 continued this work and showed that the lower bound on the domination number for trees given in Theorem 5.10 is also a lower bound on the Slater number for trees. Theorem 5.21 ([227]) If 𝑇 is a tree of order 𝑛 ≥ 3 with ℓ leaves and degree sequence (𝑑1 , 𝑑2 , . . . , 𝑑 𝑛 ), where 𝑑1 ≥ 𝑑2 ≥ · · · ≥ 𝑑 𝑛 , then sl(𝑇) ≥ 31 (𝑛 − ℓ + 2), with equality if and only if 𝑛 − ℓ ≡ 1 (mod 3) and 𝑑𝑡+1 ≤ 2, where 𝑡 = 13 (𝑛 − ℓ + 2). Corollary 5.22 ([227]) If 𝑇 is a tree of order 𝑛 ≥ 3 with ℓ leaves, where 𝑛 − ℓ ≡ 1 (mod 3) and ℓ ≤ 14 𝑛 + 2, then sl(𝑇) = 13 (𝑛 − ℓ + 2). Theorem 5.10 now follows as an immediate consequence of Theorems 5.20 and 5.21, and we can restate it as follows. Corollary 5.23 If 𝑇 is a tree of order 𝑛 with ℓ leaves and degree sequence (𝑑1 , 𝑑2 , . . . , 𝑑 𝑛 ), where 𝑑1 ≥ 𝑑2 ≥ · · · ≥ 𝑑 𝑛 , then 𝛾(𝑇) ≥ sl(𝑇) ≥ 13 (𝑛 − ℓ + 2). Further, if 𝑛 − ℓ . 1 (mod 3) or if 𝑛 − ℓ ≡ 1 (mod 3) and 𝑑𝑡+1 ≥ 3, where 𝑡 = 13 (𝑛 − ℓ + 2), then 𝛾(𝑇) > 13 (𝑛 − ℓ + 2). There exist trees for which the difference between the Slater number and 13 (𝑛−ℓ+2) is arbitrarily large. For example, let 𝑘 ≥ 3 be any odd integer, and let 𝑇 be a caterpillar with spine 𝑃 𝑘 and code (ℓ1 , ℓ2 , . . . , ℓ𝑘 ), where ℓ1 = ℓ𝑘 = 2 and ℓ𝑖 = 1 for 2 ≤ 𝑖 ≤ 𝑘 −1. Then, 𝑇 has order 𝑛 = 2𝑘 + 2, sl(𝑇) = 12 (𝑘 + 1), and 13 (𝑛 − ℓ + 2) = 13 (𝑘 + 2). Recall that R denotes the family of trees 𝑇 for which 𝑑 (𝑥, 𝑦) ≡ 2 (mod 3) between any pair of distinct leaves 𝑥 and 𝑦 of 𝑇. As mentioned earlier, Hajian et al. [372] showed that R = G00 , where G00 is the family defined in Definition 5.9. As an immediate consequence of Theorem 5.20, Corollary 5.22, and Theorem 5.10, we have the following result. Corollary 5.24 If 𝑇 is a tree of order 𝑛 ≥ 3 with ℓ leaves, where 𝑛 − ℓ ≡ 1 (mod 3) and ℓ ≤ 2 + 𝑛4 , then 𝛾(𝑇) = sl(𝑇) if and only if 𝑇 ∈ G00 . Next we present an upper bound on the domination number of a tree in terms of the Slater number. Let 𝑉≥2 (𝑇) be the set of 𝑛 − ℓ non-leaf vertices of a tree 𝑇. Theorem 5.25 ([227]) If 𝑇 is a tree of order 𝑛 ≥ 3, then 𝛾(𝑇) ≤ 3 sl(𝑇) − 2, with equality if and only if sl(𝑇) = 13 (𝑛 − ℓ + 2) and every vertex of 𝑇 is a leaf or a support vertex. Proof By Observation 5.1, 𝛾(𝑇) ≤ 𝑛 − ℓ. By Theorem 5.21, we have sl(𝑇) ≥ 1 3 (𝑛 − ℓ + 2). Hence, 𝛾(𝑇) ≤ 𝑛 − ℓ ≤ 3 sl(𝑇) − 2, (5.5)

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111

which establishes the desired upper bound. To characterize the trees achieving the upper bound, first assume that sl(𝑇) = 31 (𝑛 − ℓ + 2) and every vertex of 𝑇 is a leaf or a support vertex. Then every vertex in 𝑉≥2 (𝑇) is a support vertex of 𝑇. By Observation 5.1(a), there exists a 𝛾-set 𝐷 of 𝑇 that contains no leaf, and so 𝐷 ⊆ 𝑉≥2 (𝑇). However, in order to dominate the leaves of 𝑇, the set 𝐷 contains every support vertex of 𝑇, and so 𝑉≥2 (𝑇) ⊆ 𝐷. Consequently, 𝐷 = 𝑉≥2 (𝑇), and so 𝛾(𝑇) = |𝐷 | = 𝑛 − ℓ = 3 sl(𝑇) − 2. This proves the sufficiency. To prove the necessity, assume that 𝛾(𝑇) = 3 sl(𝑇) − 2. Then we must have equality throughout Inequality (5.5). Thus, sl(𝑇) = 13 (𝑛 − ℓ + 2) and 𝛾(𝑇) = 𝑛 − ℓ. If there is a vertex 𝑣 in 𝑉≥2 (𝑇) that is not a support vertex of 𝑇, then 𝑉≥2 (𝑇) \ {𝑣} is a dominating set of 𝑇, and so 𝛾(𝑇) ≤ |𝑉≥2 (𝑇)| − 1 = 𝑛 − ℓ − 1, a contradiction. Therefore, every vertex in 𝑉≥2 (𝑇) is a support vertex of 𝑇. For an example of a tree attaining the upper bound of Theorem 5.25, let 𝑡 ≥ 2 and let 𝑇 ′ be a caterpillar with spine 𝑣 1 𝑣 2 . . . 𝑣 𝑡 and code (ℓ1 , ℓ2 , . . . , ℓ𝑡 ), where ℓ𝑖 = 5 for 𝑖 ∈ [𝑡]. Let 𝑇 be the tree obtained from 𝑇 ′ by subdividing exactly two pendant edges incident to 𝑣 𝑖 for each 𝑖 ∈ [𝑡 − 1]. For 𝑡 = 3, the tree 𝑇 is illustrated in Figure 5.6. The resulting tree 𝑇 has 𝑛 = 8𝑡 − 2 = 22 vertices, sl(𝑇) = 𝑡 = 3, and 𝛾(𝑇) = 3𝑡 − 2 = 3 sl(𝑇) − 2 = 7. 𝑣1

𝑣2

𝑣3

Figure 5.6 A tree 𝑇 with 𝛾(𝑇) = 3 sl(𝑇) − 2 By Theorem 5.20 and Theorem 5.25, if 𝑇 is a tree of order 𝑛 ≥ 3, then sl(𝑇) ≤ 𝛾(𝑇) ≤ 3 sl(𝑇) − 2.

(5.6)

We conclude this section by showing that all values of the lower and upper bounds on 𝛾(𝑇) in Inequality (5.6) are attainable. Theorem 5.26 ([227]) For integers 𝑎 and 𝑏, such that 𝑎 ≥ 1 and 0 ≤ 𝑏 ≤ 2𝑎 − 2, there exists a tree 𝑇 for which sl(𝑇) = 𝑎 and 𝛾(𝑇) = 𝑎 + 𝑏. Proof Let 𝑎 and 𝑏 be integers where 𝑎 ≥ 1 and 0 ≤ 𝑏 ≤ 2𝑎 − 2. In the case when 𝑎 = 1, we have that 𝑏 = 0, and we take 𝑇 to be a star on at least three vertices. Then, sl(𝑇) = 1 = 𝑎 and 𝛾(𝑇) = 1 = 𝑎 + 𝑏. Assume that 𝑎 ≥ 2. Begin with the caterpillar 𝑇 ′ having spine 𝑣 1 𝑣 2 . . . 𝑣 𝑎 and code (ℓ1 , ℓ2 , . . . , ℓ𝑎 ), where ℓ𝑖 = 2𝑎 for 1 ≤ 𝑖 ≤ 𝑎. Let ′ 𝑅 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑎 }. If 𝑏 is even, let 𝑇 be the tree obtained  𝑏  from 𝑇 by subdividing exactly two pendant edges incident to 𝑣 𝑖 for each 𝑖 ∈ 2 . If 𝑏 is odd, let 𝑇 be the tree obtained from 𝑇 ′ by subdividing exactly two pendant edges incident to 𝑣 𝑖 for

112

Chapter 5. Domination in Trees

  each 𝑖 ∈ 𝑏−1 2 , and subdividing exactly one pendant edge incident to 𝑣 (𝑏+1)/2 . Thus, the resulting tree 𝑇 has order 𝑛 = 2𝑎 2 + 𝑎 + 𝑏. It follows that ∑︁ |𝑅| + deg(𝑣) = |𝑅| + (2𝑎 + 2)|𝑅| − 2 𝑣∈𝑅 = (2𝑎 + 3)|𝑅| − 2 = (2𝑎 + 3)𝑎 − 2 = 2𝑎 2 + 3𝑎 − 2 ≥ 2𝑎 2 + 𝑎 + 𝑏 = 𝑛, implying that sl(𝑇) ≤ |𝑅| = 𝑎. However, if 𝑅 ′ = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑎−1 }, then 𝑅 ′ is a set of 𝑎 − 1 vertices of largest possible degree and ∑︁ |𝑅 ′ | + deg(𝑣) = |𝑅 ′ | + (2𝑎 + 2)|𝑅 ′ | − 1 ′ 𝑣 ∈𝑅 = (2𝑎 + 3)|𝑅 ′ | − 1 = (2𝑎 + 3) (𝑎 − 1) − 1 = 2𝑎 2 + 𝑎 − 4 < 𝑛, implying that sl(𝑇) > |𝑅 ′ | = 𝑎 − 1. Consequently, sl(𝑇) = 𝑎. We note that every vertex of 𝑇 is a leaf or a support vertex and since 𝑇 has 𝑎 + 𝑏 support vertices, every one of which must be in every 𝛾-set of 𝑇, it follows that 𝛾(𝑇) = 𝑎 + 𝑏.

5.2.4

Vertices in All or No Minimum Dominating Sets

A vertex can appear in every 𝛾-set, in no 𝛾-set, or in some but not all 𝛾-sets of a graph. For example, the double star 𝑇 = 𝑆(1, 4) with centers 𝑥 and 𝑦, where 𝑥 is adjacent to one leaf 𝑥 ′ and 𝑦 is adjacent to four leaves, has exactly two 𝛾-sets. The vertex 𝑦 is in both 𝛾-sets of 𝑇, each of the four leaves adjacent to 𝑦 is in no 𝛾-set of 𝑇, and each of 𝑥 and 𝑥 ′ appears in exactly one of the two 𝛾-sets of 𝑇. In this section, we are concerned with the vertices in all or no minimum dominating sets of a tree 𝑇. Let these sets be defined as follows: • A (𝐺) = 𝑣 ∈ 𝑉 (𝐺) : 𝑣 is in all 𝛾-sets of 𝐺 , and  • N (𝐺) = 𝑣 ∈ 𝑉 (𝐺) : 𝑣 is in no 𝛾-set of 𝐺 . For example,  the path 𝑃𝑛 given by 𝑣 1 𝑣 2 . . . 𝑣 𝑛−1 𝑣 𝑛 , where 𝑛 ≡ 0 (mod 3), has A (𝑃𝑛 ) = 𝑣 𝑖 : 𝑖 ≡ 2 (mod 3) and 𝑖 ∈ [𝑛] and N (𝑃𝑛 ) = 𝑣 𝑖 : 𝑖 ≡ 0, 1 (mod 3) and 𝑖 ∈ [𝑛] . Mynhardt [604] in 1999 introduced a novel technique called tree pruning to help characterize the sets A (𝑇) and N (𝑇) for an arbitrary tree 𝑇. Variations of this technique have been used to characterize vertices contained in all, or in no, 𝜇-sets of trees for other domination parameters 𝜇. For examples, see [80, 197, 470, 474]. We give a slightly modified description of Mynhardt’s pruning technique here. A branch vertex is a vertex of degree at least 3 in 𝑇. Let 𝐿 (𝑇) denote the set of leaves

Section 5.2. Domination in Trees

113

of 𝑇. We denote the set of leaves in the rooted tree 𝑇 = 𝑇𝑣 distinct from 𝑣 by 𝐿(𝑣), that is, 𝐿(𝑣) = 𝐷 (𝑣) ∩ 𝐿 (𝑇). For 𝑗 ∈ [2], we define  𝐿 𝑗 (𝑣) = 𝑢 ∈ 𝐿 (𝑣) : 𝑑 (𝑢, 𝑣) ≡ 𝑗 (mod 3) . The pruning of a tree is performed with respect to the root. Hence, suppose 𝑇 is rooted at 𝑣, that is, 𝑇 = 𝑇𝑣 . If deg(𝑢) ≤ 2 for each 𝑢 ∈ 𝑉 (𝑇𝑣 ) \ {𝑣}, then let 𝑇 𝑣 = 𝑇. Otherwise, let 𝑢 be a branch vertex at maximum distance from 𝑣. Note that |𝐶 (𝑢)| ≥ 2 and deg(𝑥) ≤ 2 for each 𝑥 ∈ 𝐷 (𝑢). The pruning process is applied as follows: • If |𝐿 1 (𝑢)| ≥ 1, then delete 𝐷 (𝑢) and attach a new leaf adjacent to 𝑢. • If 𝐿 1 (𝑢) = ∅ and |𝐿 2 (𝑢)| ≥ 1, then delete 𝐷 (𝑢) and attach a path of length 2 to 𝑢. • If 𝐿 1 (𝑢) = 𝐿 2 (𝑢) = ∅, then delete 𝐷 (𝑢) and attach a path of length 3 to 𝑢. This step of the pruning process, where all the descendants of 𝑢 are deleted and a path of length 1, 2, or 3 is attached to 𝑢 to give a tree in which 𝑢 has degree 2, is called a pruning of 𝑇𝑣 at 𝑢. Repeat the above process until a tree 𝑇 𝑣 is obtained with deg(𝑢) ≤ 2 for each 𝑢 ∈ 𝑉 (𝑇 𝑣 ) \ {𝑣}. Then, 𝑇 𝑣 is called a pruning of 𝑇𝑣 . The tree 𝑇 𝑣 is unique. We note that if 𝑇 is a star 𝐾1,𝑛−1 for 𝑛 ≥ 3 rooted at its center 𝑣, then 𝑇 = 𝑇 𝑣 . To illustrate the pruning process, consider the tree 𝑇𝑣 in Figure 5.7(a). The vertices 𝑢, 𝑤, and 𝑥 are branch vertices at maximum distance 2 from 𝑣. Since 𝐿 1 (𝑢) = ∅ and |𝐿 2 (𝑢)| = 1, we delete 𝐷 (𝑢) and attach a path of length 2 to 𝑢. Since |𝐿 1 (𝑤)| = 2, we delete 𝐷 (𝑤) and attach a path of length 1 to 𝑤. Similarly, since |𝐿 1 (𝑥)| = 2, we delete 𝐷 (𝑥) and attach a path of length 1 to 𝑥. This pruning of 𝑇𝑣 at 𝑢, 𝑤, and 𝑥 produces the intermediate tree 𝑇𝑣′ shown in Figure 5.7(b). In tree 𝑇𝑣′ , the vertices 𝑦 and 𝑧 are branch vertices at maximum distance 1 from 𝑣. Since 𝐿 1 (𝑦) = 𝐿 2 (𝑦) = ∅, we delete 𝐷 (𝑦) and attach a path of length 3 to 𝑦. Since |𝐿 1 (𝑧)| = 1, we delete 𝐷 (𝑧) and attach a path of length 1 to 𝑧. This pruning of 𝑇𝑣′ at 𝑦 and 𝑧 produces the pruning 𝑇 𝑣 of 𝑇𝑣 . The following characterization of the sets A (𝑇) and N (𝑇) for an arbitrary tree 𝑇 is a slightly modified version of the one presented in [604]. Theorem 5.27 ([604]) If 𝑣 is a vertex of a tree 𝑇 and 𝐿 𝑖 (𝑣) equals 𝐿 𝑖 (𝑣) in the pruning 𝑇 𝑣 of 𝑇𝑣 , then the following hold: (a) 𝑣 ∈ A (𝑇) if and only if |𝐿 1 (𝑣)| ≥ 2. (b) 𝑣 ∈ N (𝑇) if and only if 𝐿 1 (𝑣) = ∅ and 𝐿 2 (𝑣) ≠ ∅. To illustrate Theorem 5.27, note that in the pruning 𝑇 𝑣 of the tree 𝑇𝑣 in Figure 5.7(a), |𝐿 0 (𝑣)| = 1, |𝐿 1 (𝑣)| = 3, and |𝐿 2 (𝑣)| = 2. Since |𝐿 1 (𝑣)| ≥ 2, 𝑣 ∈ A (𝑇) by Theorem 5.27. We conclude this section by briefly mentioning a couple of related topics. Note that if a graph has a unique minimum dominating set, then each vertex either belongs to the unique 𝛾-set or to no 𝛾-set. Gunther et al. [369] in 1994 characterized the trees having a unique 𝛾-set as follows. We note that Fischermann [300] generalized this result to block graphs in 2001.

Chapter 5. Domination in Trees

114 𝑣

𝑣

𝑦

𝑧

𝑢

𝑤

𝑦 𝑥

𝑧

𝑢

𝑤

𝑥

(b) 𝑇𝑣′

(a) 𝑇𝑣

𝑣

𝑦

𝑧

𝑢

𝑤

(c) 𝑇 𝑣

Figure 5.7 The pruning 𝑇 𝑣 of the tree 𝑇𝑣

Theorem 5.28 ([369]) If 𝑇 is a tree of order 𝑛 ≥ 3, then the following conditions are equivalent: (a) 𝑇 has a unique 𝛾-set. (b) 𝑇 has a 𝛾-set 𝑆 for which every vertex in 𝑆 has at least two 𝑆-external private neighbors. (c) 𝑇 has a 𝛾-set 𝑆 for which every vertex 𝑥 ∈ 𝑆 has the property 𝛾(𝑇 − 𝑥) > 𝛾(𝑇). Considering the vertices of a graph that appear in at least one 𝛾-set, in 2002 Fricke et al. [311] defined a partition of the vertices of a graph 𝐺 into two sets as follows. A vertex is called 𝛾-good if it is contained in some 𝛾-set of 𝐺 and 𝛾-bad, otherwise. Further, a graph 𝐺 is called 𝛾-excellent if every vertex of 𝐺 is 𝛾-good. Terminology for good, bad, and excellent is defined similarly for other domination parameters. The families of 𝛾-excellent trees, 𝛾t -excellent trees, and 𝑖-excellent trees were also studied in [127], [420], and [456], respectively.

5.2.5

Domination and Packing in Trees

Recall that a packing in a graph 𝐺 is a set of vertices whose closed neighborhoods are pairwise disjoint, and the packing number 𝜌(𝐺) is the maximum cardinality of a packing in 𝐺. In Theorem 4.45 of Chapter 4, we showed that if 𝐺 is a graph, then 𝛾(𝐺) ≥ 𝜌(𝐺). That is, the packing number is a lower bound on the domination number for general graphs. In 1975 Meir and Moon [589] proved that these numbers are equal for trees.

Section 5.3. Total Domination in Trees

115

Theorem 5.29 (Meir-Moon Theorem [589]) For every tree 𝑇, 𝜌(𝑇) = 𝛾(𝑇). Proof We proceed by induction on the order 𝑛 of a tree 𝑇 to show that 𝜌(𝑇) = 𝛾(𝑇). If 𝑛 ≤ 3, then 𝜌(𝑇) = 𝛾(𝑇) = 1. This establishes the base cases. Let 𝑛 ≥ 4 and assume that if 𝑇 ′ is a tree of order 𝑛′ , where 1 ≤ 𝑛′ < 𝑛, then 𝜌(𝑇 ′ ) = 𝛾(𝑇 ′ ). Let 𝑇 be a tree of order 𝑛. By Theorem 4.45, we have 𝛾(𝑇) ≥ 𝜌(𝑇). If diam(𝑇) = 2, then 𝑇 is a star and 𝜌(𝑇) = 𝛾(𝑇) = 1. If diam(𝑇) = 3, then 𝑇 is a double star and 𝜌(𝑇) = 𝛾(𝑇) = 2. Hence, we may assume that diam(𝑇) ≥ 4, and so 𝑛 ≥ 5, for otherwise the desired result follows. Let 𝑣 0 𝑣 1 . . . 𝑣 𝑑 be a longest path in 𝑇, and so 𝑑 = diam(𝑇) ≥ 4. We now root the tree 𝑇 at the vertex 𝑟 = 𝑣 𝑑 . Suppose that deg𝑇 (𝑣 2 ) = 2. In this case, we let 𝑇 ′ = 𝑇 − 𝑉 (𝑇𝑣2 ), where 𝑇𝑣2 is the maximal subtree of 𝑇 rooted at 𝑣 2 . The resulting tree 𝑇 ′ has order 𝑛′ , where 2 ≤ 𝑛′ ≤ 𝑛 − 3. Every 𝛾-set of 𝑇 ′ can be extended to a dominating set of 𝑇 by adding to it the vertex 𝑣 1 , and so 𝛾(𝑇) ≤ 𝛾(𝑇 ′ ) + 1. Every maximum packing in 𝑇 ′ can be extended to a packing in 𝑇 by adding to it the vertex 𝑣 0 , and so 𝜌(𝑇) ≥ 𝜌(𝑇 ′ ) + 1. Applying the inductive hypothesis to 𝑇 ′ , we have that 𝜌(𝑇 ′ ) = 𝛾(𝑇 ′ ). Therefore, 𝜌(𝑇) ≥ 𝜌(𝑇 ′ ) + 1 = 𝛾(𝑇 ′ ) + 1 ≥ 𝛾(𝑇) ≥ 𝜌(𝑇).

(5.7)

Consequently, we must have equality throughout Inequality (5.7). In particular, 𝛾(𝑇) = 𝜌(𝑇). Hence, we may assume that deg𝑇 (𝑣 2 ) ≥ 3, for otherwise the desired result holds. We now consider the tree 𝑇 ′ = 𝑇 − 𝑉 (𝑇𝑣1 ), where 𝑇𝑣1 is the maximal subtree of 𝑇 rooted at 𝑣 1 . The resulting tree 𝑇 ′ has order 𝑛′ , where 4 ≤ 𝑛′ ≤ 𝑛 − 2. Let 𝑆 be a maximum packing in 𝑇 ′ , and so 𝜌(𝑇 ′ ) = |𝑆|. If 𝑣 2 ∈ 𝑆, then since every descendant of 𝑣 2 is at distance 1 or 2 from 𝑣 2 , we note that no descendant of 𝑣 2 belongs to the set 𝑆. In this case, if 𝑢 1 is an arbitrary child of 𝑣 2 different from 𝑣 1 , then we can replace the vertex 𝑣 2 in 𝑆 with the vertex 𝑢 1 to yield a new maximum packing in 𝑇 ′ . Hence, we can choose the maximum packing 𝑆 in 𝑇 ′ so that 𝑣 2 ∉ 𝑆. With this choice of 𝑆, the set 𝑆 ∪ {𝑣 0 } is a packing in 𝑇, implying that 𝜌(𝑇) ≥ |𝑆| + 1 = 𝜌(𝑇 ′ ) + 1. Applying the inductive hypothesis to 𝑇 ′ , we have that 𝜌(𝑇 ′ ) = 𝛾(𝑇 ′ ). Furthermore, every 𝛾-set of 𝑇 ′ can be extended to a dominating set of 𝑇 by adding vertex 𝑣 1 , and so 𝛾(𝑇) ≤ 𝛾(𝑇 ′ ) + 1. Therefore, once again Inequality (5.7) holds, implying that 𝛾(𝑇) = 𝜌(𝑇).

5.3

Total Domination in Trees

In this section, we present several upper and lower bounds on the total domination number of a tree, we present a discussion of vertices in all or no 𝛾t -sets in trees, and we discuss trees having unique 𝛾t -sets. We also present Rall’s theorem that for all trees, the total domination number equals the open packing number.

5.3.1

Total Domination Bounds in Trees

Recall that in Section 4.3.2, we presented the result due to Cockayne et al. [182] showing that the total domination of a connected graph of order at least 3 is at most

Chapter 5. Domination in Trees

116

two-thirds its order. We also presented the characterization due to Brigham et al. [117] of the graphs that achieve equality in this upper bound. Here we restate these results in the special case of trees. Theorem 5.30 ([117, 182]) If 𝑇 is tree of order 𝑛 ≥ 3, then 𝛾(𝑇) ≤ equality if and only if 𝑇 = 𝑇 ′ ◦ 𝑃2 is the 2-corona of some tree 𝑇 ′ .

2 3 𝑛,

with

5.3.2 Total Domination Bounds Involving the Number of Leaves In 2006 Chellali and Haynes [153] were the first to establish a lower bound on the total domination number of a tree in terms of the order, number of leaves, and number of support vertices in the tree. In order to state their result, we define the family F0 of trees as follows. Definition 5.31 Let F0 be the family of trees 𝑇 that can be obtained from a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 , for 𝑘 ≥ 1, of trees, where the tree 𝑇1 is the path 𝑃4 with support vertices 𝑥 and 𝑦, and where the tree 𝑇 = 𝑇𝑘 . Further, if 𝑘 ≥ 2, then for each 𝑖, where 2 ≤ 𝑖 ≤ 𝑘, the tree 𝑇𝑖 can be obtained from the tree 𝑇𝑖−1 by applying one of the Operations O1 , O2 , and O3 defined as follows. Let 𝐴(𝑇1 ) = {𝑥, 𝑦} and 𝐻 be a path 𝑃4 with non-leaf vertices labeled 𝑢 and 𝑣. Operation O1 . Add a new vertex to 𝑇𝑖−1 and join it to a vertex of 𝐴(𝑇𝑖−1 ). Let 𝐴(𝑇𝑖 ) = 𝐴(𝑇𝑖−1 ). Operation O2 . Add a copy of 𝐻 to 𝑇𝑖−1 and add an edge from a leaf of 𝐻 to a leaf of 𝑇𝑖−1 . Let 𝐴(𝑇𝑖 ) = 𝐴(𝑇𝑖−1 ) ∪ {𝑢, 𝑣}. Operation O3 . Add a copy of 𝐻 to 𝑇𝑖−1 and add a new vertex 𝑤, an edge from 𝑤 to a support vertex 𝑢 of 𝐻, and an edge from 𝑤 to a leaf of 𝑇𝑖−1 . Let 𝐴(𝑇𝑖 ) = 𝐴(𝑇𝑖−1 ) ∪ {𝑢, 𝑣}. We are now in a position to state the result from [153], albeit without proof. Theorem 5.32 ([153]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then   𝑛−ℓ+2 𝛾t (𝑇) ≥ , 2 with equality if and only if 𝑇 ∈ F0 . Theorem 5.32 was generalized by Hajian et al. [373] in 2019, who defined the families F01 , F02 , F03 , and F0★ of trees as follows. Definition 5.33 Let F01 , F02 , and F03 be the families of trees defined as follows. • Let F01 be the family of trees 𝑇 that can be obtained from a tree 𝑇 ′ ∈ F0 by adding 𝑎 ≥ 1 new vertices and joining all of them to the same leaf of 𝑇 ′ . • Let F02 be the family of trees 𝑇 that can be obtained from a tree 𝑇 ′ ∈ F0 that contains a support vertex 𝑥 all of whose neighbors, except for exactly one neighbor, are leaves in 𝑇 ′ by removing all leaf neighbors of 𝑥. • Let F03 be the family of trees 𝑇 that can be obtained from a tree 𝑇 ′ ∈ F0 by adding a double star 𝑄 and adding an edge from a leaf of 𝑄 to a vertex of degree at least 2 in 𝑇 ′ .

Section 5.3. Total Domination in Trees

117

Definition 5.34 Let F0★ be the family of trees 𝑇 that can be obtained from a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 of trees, where 𝑘 ≥ 1 and where the tree 𝑇1 ∈ F01 ∪ F02 ∪ F03 and the tree 𝑇 = 𝑇𝑘 . Further, if 𝑘 ≥ 2, then for each 𝑖, where 2 ≤ 𝑖 ≤ 𝑘, the tree 𝑇𝑖 can be obtained from the tree 𝑇𝑖−1 by applying Operation O★ defined below. Operation O★. Add a double star 𝑄 to 𝑇𝑖−1 and add an edge between a leaf of 𝑄 and a leaf of 𝑇𝑖−1 . The following result generalizes Theorem 5.32. Theorem 5.35 ([373]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then   𝑛−ℓ+2 𝛾t (𝑇) ≥ , 2 with equality if and only if 𝑇 ∈ F0 ∪ F0★. We remark that in [373], Theorem 5.35 for trees was generalized to show that if 𝐺 is aconnected graph of order 𝑛 ≥ 2 with 𝑘 ≥ 0 cycles and ℓ leaves, then  𝛾t (𝐺) ≥ 12 (𝑛 − ℓ + 2) − 𝑘. The graphs 𝐺 that achieve equality for this generalized bound were also characterized in [373]. In 2004 Chellali and Haynes [152] established the following upper bound on the total domination number of a tree in terms of the number of support vertices. Theorem 5.36 ([152]) If 𝑇 is a tree of order 𝑛 ≥ 3 with 𝑠 support vertices, then 𝛾t (𝑇) ≤ 12 (𝑛 + 𝑠). In 2015 Krzywkowski [548] gave an algorithmic procedure to build trees attaining the upper bound of Theorem 5.36. Since the number of support vertices in a tree is at most the number of leaves in the tree, the following is an immediate consequence of Theorem 5.36. Corollary 5.37 ([152, 153]) If 𝑇 is a tree of order 𝑛 ≥ 3 with ℓ leaves, then 𝛾t (𝑇) ≤ 12 (𝑛 + ℓ). A characterization of the trees achieving equality in the bound of Corollary 5.37 can be found in the book [490]. Henning and Yeo [490] stated that this characterization was adapted from a result due to Chen and Sohn [156].

5.3.3

Vertices in All or No Minimum Total Dominating Sets in Trees

In this section, we are concerned with the vertices in all or no minimum total dominating sets. Unless otherwise stated, we use the same notation here as in the previous section.  Let these sets be defined as follows: • A𝑡 (𝐺) = 𝑣 ∈ 𝑉 (𝐺) : 𝑣 is in every 𝛾t -set of 𝐺 , and  • N𝑡 (𝐺) = 𝑣 ∈ 𝑉 (𝐺) : 𝑣 is in no 𝛾t -set of 𝐺 .

Chapter 5. Domination in Trees

118 For 𝑗 ∈ [3] 0 , we define

 𝐿 𝑗 (𝑣) = 𝑢 ∈ 𝐿 (𝑣) : 𝑑 (𝑢, 𝑣) ≡ 𝑗 (mod 4) . Again, the pruning of a tree is performed with respect to the root. For total domination, the pruning process is as follows: • If |𝐿 2 (𝑢)| ≥ 1, then delete 𝐷 (𝑢) and attach a path of length 2 to 𝑢. • If |𝐿 1 (𝑢)| ≥ 1, 𝐿 2 (𝑢) = ∅, and |𝐿 3 (𝑢)| ≥ 1, then delete 𝐷 (𝑢) and attach a path of length 2 to 𝑢. • If |𝐿 1 (𝑢)| ≥ 1 and 𝐿 2 (𝑢) = 𝐿 3 (𝑢) = ∅, then delete 𝐷 (𝑢) and attach a path of length 1 to 𝑢. • If 𝐿 1 (𝑢) = 𝐿 2 (𝑢) = ∅ and |𝐿 3 (𝑢)| ≥ 1, then delete 𝐷 (𝑢) and attach a path of length 3 to 𝑢. • If 𝐿 1 (𝑢) = 𝐿 2 (𝑢) = 𝐿 3 (𝑢) = ∅, then delete 𝐷 (𝑢) and attach a path of length 4 to 𝑢. The steps of the pruning process are repeated as before until a tree 𝑇 𝑣 is obtained with deg(𝑢) ≤ 2 for each 𝑢 ∈ 𝑉 (𝑇 𝑣 ) \ {𝑣}. To illustrate the pruning process for total domination, consider the tree 𝑇𝑣 in Figure 5.8(a). The vertices 𝑢 and 𝑤 are branch vertices at maximum distance 2 from 𝑣. Since |𝐿 2 (𝑢)| = 1, we delete 𝐷 (𝑢) and attach a path of length 2 to 𝑢. Since |𝐿 1 (𝑤)| = 2 and 𝐿 2 (𝑤) = 𝐿 3 (𝑤) = ∅, we delete 𝐷 (𝑤) and attach a path of length 1 to 𝑤. This pruning of 𝑇𝑣 at 𝑢 and 𝑤 produces the intermediate tree 𝑇𝑣′ shown in Figure 5.8(b). In tree 𝑇𝑣′ , the vertices 𝑡, 𝑥, 𝑦, and 𝑧 are branch vertices at maximum distance 1 from 𝑣. Since |𝐿 2 (𝑥)| = 1, we delete 𝐷 (𝑥) and attach a path of length 2 to 𝑥. Since |𝐿 1 (𝑦)| = 1, 𝐿 2 (𝑦) = ∅, and |𝐿 3 (𝑦)| = 1, we delete 𝐷 (𝑦) and attach a path of length 2 to 𝑦. Since 𝐿 1 (𝑡) = 𝐿 2 (𝑡) = 𝐿 3 (𝑡) = ∅, we delete 𝐷 (𝑡) and attach a path of length 4 to 𝑡. Since 𝐿 1 (𝑧) = 𝐿 2 (𝑧) = ∅ and |𝐿 3 (𝑧)| = 2 ≥ 1, we delete 𝐷 (𝑧) and attach a path of length 3 to 𝑧. This pruning of 𝑇𝑣′ at 𝑡, 𝑥, 𝑦, and 𝑧 produces the pruning 𝑇 𝑣 of 𝑇𝑣 . The following characterization of the sets A𝑡 (𝑇) and N𝑡 (𝑇) for an arbitrary tree 𝑇 is given by Cockayne et al. [197] in 2003. Theorem 5.38 ([197]) If 𝑣 is a vertex of a tree 𝑇 and 𝐿 𝑖 (𝑣) equals 𝐿 𝑖 (𝑣) in the pruning 𝑇 𝑣 of 𝑇𝑣 , then the following hold: (a) 𝑣 ∈ A𝑡 (𝑇) if and only if 𝑣 is a support vertex or |𝐿 1 (𝑣) ∪ 𝐿 2 (𝑣)| ≥ 2. (b) 𝑣 ∈ N𝑡 (𝑇) if and only if 𝐿 1 (𝑣) ∪ 𝐿 2 (𝑣) = ∅. To illustrate Theorem 5.38, note that in the pruning 𝑇 𝑣 of the tree 𝑇𝑣 in Figure 5.8(a), |𝐿 0 (𝑣)| = |𝐿 1 (𝑣)| = |𝐿 2 (𝑣)| = 1 and |𝐿 3 (𝑣)| = 2, that is, |𝐿 1 (𝑣) ∪ 𝐿 2 (𝑣)| = 2. Hence, by Theorem 5.38, we have 𝑣 ∈ A𝑡 (𝑇).

5.3.4

Unique Minimum Total Dominating Sets in Trees

We now turn our attention to trees having a unique 𝛾t -set. We need to recall the following terminology from Section 5.2.4. As before, we denote the set of leaves in

Section 5.3. Total Domination in Trees

119

𝑣

𝑦

𝑡

𝑧

𝑣

𝑥

𝑢

𝑦 𝑤

𝑡

𝑧

𝑢

𝑥 𝑤

(b) 𝑇𝑣′

(a) 𝑇𝑣

𝑣

𝑦

𝑡

𝑧

𝑥

(c) 𝑇 𝑣

Figure 5.8 The pruning 𝑇 𝑣 of the tree 𝑇𝑣

𝑇 = 𝑇𝑣 distinct from 𝑣 by 𝐿 (𝑣) that is, 𝐿 (𝑣) = 𝐷 (𝑣) ∩ 𝐿(𝑇), where 𝐿(𝑇) is the set of leaves of 𝑇. For 𝑗 ∈ [3] 0 , let  𝐿 𝑗 (𝑣) = 𝑢 ∈ 𝐿(𝑣) : 𝑑 (𝑢, 𝑣) ≡ 𝑗 (mod 4) , and let 𝐿 𝑗 (𝑣) equal 𝐿 𝑗 (𝑣) in the pruning 𝑇 𝑣 of 𝑇𝑣 . In 2002 Haynes and Henning [421] characterized trees that have a unique 𝛾t -set as follows. Theorem 5.39 ([421]) If 𝑇 is a nontrivial tree, then the following conditions are equivalent: (a) 𝑇 has a unique 𝛾t -set. (b) 𝑇 has a 𝛾t -set 𝑆 for which every vertex 𝑣 ∈ 𝑆 is either a support vertex or satisfies |pn[𝑣, 𝑆] | ≥ 2. (c) 𝑇 has a 𝛾t -set 𝑆 for which 𝛾t (𝑇 − 𝑣) > 𝛾t (𝑇) for every 𝑣 ∈ 𝑆 that is not a support vertex. (d) For every vertex 𝑣 ∈ 𝑉 (𝑇), 𝑣 is a support vertex or |𝐿 1 (𝑣) ∪ 𝐿 2 (𝑣)| ≠ 1. In addition to providing the three equivalent conditions of Theorem 5.39 for a tree to have a unique 𝛾t -set, Haynes and Henning [421] gave a constructive characterization of such trees. We note that these trees were also independently characterized by Fischermann in [301] in 2004.

120

Chapter 5. Domination in Trees

5.3.5 Total Domination and Open Packing in Trees Recall that an open packing in a graph 𝐺 is a set of vertices whose open neighborhoods are pairwise disjoint. Thus, if 𝑆 is an open packing in 𝐺, then N(𝑢) ∩ N(𝑣) = ∅ for all 𝑢, 𝑣 ∈ 𝑆. The open packing number 𝜌 o (𝐺) is the maximum cardinality of an open packing in 𝐺. Recall also that in Theorem 4.46 in Chapter 4, we showed that if 𝐺 is an isolate-free graph, then the open packing number is a lower bound on the total domination number of 𝐺. We restate this result for trees. Proposition 5.40 If 𝑇 is a nontrivial tree, then 𝜌 o (𝑇) ≤ 𝛾t (𝑇). In 2005 Rall [644] showed that 𝜌 o (𝑇) = 𝛾t (𝑇) for all nontrivial trees 𝑇, using an elegant proof involving categorical products. The proof we present uses elementary properties of a tree along similar lines to the proof of Theorem 5.29 showing that the domination and packing numbers are equal for trees. We first observe the following elementary properties of a TD-set in a graph. Observation 5.41 The following properties hold in an isolate-free graph 𝐺: (a) Every TD-set in 𝐺 contains the set of support vertices of 𝐺. (b) If 𝐺 is connected and diam(𝐺) ≥ 3, then there exists a 𝛾t -set of 𝐺 that contains no leaf of 𝐺. Theorem 5.42 ([644]) If 𝑇 is a nontrivial tree, then 𝜌 o (𝑇) = 𝛾t (𝑇). Proof We proceed by induction on the order 𝑛 ≥ 2 of a tree 𝑇 to show that 𝜌 o (𝑇) = 𝛾t (𝑇). If 𝑛 ≤ 4, then 𝜌 o (𝑇) = 2 = 𝛾t (𝑇). This establishes the base cases. Let 𝑛 ≥ 5 and assume that if 𝑇 ′ is a tree of order 𝑛′ , where 2 ≤ 𝑛′ < 𝑛, then 𝜌 o (𝑇 ′ ) = 𝛾(𝑇 ′ ). Let 𝑇 be a tree of order 𝑛. Suppose that 𝑇 contains a strong support vertex 𝑣. Let 𝑢 and 𝑤 be two leaf neighbors of 𝑣, and consider the tree 𝑇 ′ = 𝑇 − 𝑢. Every TD-set of 𝑇 ′ contains the support vertex 𝑣, and is therefore a TD-set of 𝑇, implying that 𝛾t (𝑇) ≤ 𝛾t (𝑇 ′ ). Every maximum open packing in 𝑇 ′ is an open packing in 𝑇, implying that 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ). By Proposition 5.40, we have 𝛾t (𝑇) ≥ 𝜌 o (𝑇), and applying the inductive hypothesis to 𝑇 ′ , we have 𝜌 o (𝑇 ′ ) = 𝛾t (𝑇 ′ ). These observations imply that 𝛾t (𝑇) ≥ 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ) = 𝛾t (𝑇 ′ ) ≥ 𝛾t (𝑇).

(5.8)

Consequently, we must have equality throughout Inequality (5.8). In particular, 𝜌 o (𝑇) = 𝛾t (𝑇). Hence, we may assume that 𝑇 contains no strong support vertices, for otherwise the equality follows. With this assumption and since 𝑛 ≥ 5, we note that diam(𝑇) ≥ 4. Let 𝑣 0 𝑣 1 . . . 𝑣 𝑑 be a longest path in 𝑇, and so 𝑑 = diam(𝑇) ≥ 4. We now root the tree 𝑇 at the vertex 𝑟 = 𝑣 𝑑 . Since the vertex 𝑣 1 is not a strong support vertex, we have deg𝑇 (𝑣 1 ) = 2. Suppose that 𝑣 2 is a support vertex. Let 𝑢 be the (unique) leaf neighbor of 𝑣 2 , and consider the tree 𝑇 ′ = 𝑇 − 𝑢. By Observation 5.41, there is a 𝛾t -set 𝐷 ′ of 𝑇 ′ that contains no leaf of 𝑇 ′ , implying that {𝑣 1 , 𝑣 2 } ⊆ 𝐷 ′ . The set 𝐷 ′ is therefore a TD-set of 𝑇, and so 𝛾t (𝑇) ≤ |𝐷 ′ | = 𝛾t (𝑇 ′ ). Every maximum open packing in 𝑇 ′ is an open packing in 𝑇, implying that 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ). Applying the inductive hypothesis to 𝑇 ′ ,

Section 5.3. Total Domination in Trees

121

we have 𝜌 o (𝑇 ′ ) = 𝛾t (𝑇 ′ ). By Proposition 5.40, we have 𝛾t (𝑇) ≥ 𝜌 o (𝑇). Therefore, once again equality in Inequality (5.8) holds, implying that 𝜌 o (𝑇) = 𝛾t (𝑇). Hence, we may assume that 𝑣 2 is not a support vertex, for otherwise the desired result holds. Hence, every child of 𝑣 2 is a support vertex of degree 2 in 𝑇. Suppose that deg𝑇 (𝑣 2 ) ≥ 3. Let 𝑢 1 be a child of 𝑣 2 different from 𝑣 1 , and let 𝑢 0 be the child of 𝑢 1 . We now consider the tree 𝑇 ′ = 𝑇 − {𝑢 0 , 𝑢 1 }. By Observation 5.41, we can choose a 𝛾t -set 𝐷 ′ of 𝑇 ′ so that {𝑣 1 , 𝑣 2 } ⊆ 𝐷 ′ . Since the set 𝐷 ′ ∪ {𝑢 1 } is a TD-set of 𝑇, we have 𝛾t (𝑇) ≤ |𝐷 ′ | + 1 = 𝛾t (𝑇 ′ ) + 1. Let 𝑆 be a maximum open packing in 𝑇 ′ , and so 𝜌 o (𝑇 ′ ) = |𝑆|. If 𝑣 2 ∈ 𝑆, then 𝑣 0 does not belong to 𝑆. In this case, we can replace the vertex 𝑣 2 in 𝑆 with the vertex 𝑣 0 to yield a new maximum open packing in 𝑇 ′ . Hence, we can choose the maximum open packing 𝑆 in 𝑇 ′ so that 𝑣 2 ∉ 𝑆. With this choice of 𝑆, the set 𝑆 ∪ {𝑢 0 } is an open packing in 𝑇, implying that 𝜌 o (𝑇) ≥ |𝑆| + 1 = 𝜌 o (𝑇 ′ ) + 1. Applying the inductive hypothesis to 𝑇 ′ , we have 𝜌 o (𝑇 ′ ) = 𝛾t (𝑇 ′ ). By Proposition 5.40, we have 𝛾t (𝑇) ≥ 𝜌 o (𝑇). These observations imply that (5.9) 𝛾t (𝑇) ≥ 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ) + 1 = 𝛾t (𝑇 ′ ) + 1 ≥ 𝛾t (𝑇). Consequently, we must have equality throughout Inequality (5.9). In particular, 𝜌 o (𝑇) = 𝛾t (𝑇). Hence, we may assume that deg𝑇 (𝑣 2 ) = 2. Suppose that deg𝑇 (𝑣 3 ) = 2. In this case, if 𝑛 = 5, then 𝑇 = 𝑃5 and 𝜌 o (𝑇) = 𝛾(𝑇) = 3. Hence, we may assume that 𝑛 ≥ 6. We now consider the tree 𝑇 ′ = 𝑇 − {𝑣 0 , 𝑣 1 , 𝑣 2 , 𝑣 3 }. Every 𝛾t -set of 𝑇 ′ can be extended to a TD-set of 𝑇 by adding to it the vertices 𝑣 1 and 𝑣 2 , implying that 𝛾t (𝑇) ≤ 𝛾t (𝑇 ′ ) + 2. Every maximum open packing in 𝑇 ′ can be extended to an open packing in 𝑇 by adding to it the vertices 𝑣 0 and 𝑣 1 , implying that 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ) + 2. Applying the inductive hypothesis to 𝑇 ′ , we have 𝜌 o (𝑇 ′ ) = 𝛾t (𝑇 ′ ). By Proposition 5.40, we have 𝛾t (𝑇) ≥ 𝜌 o (𝑇). These observations imply that (5.10) 𝛾t (𝑇) ≥ 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ) + 2 = 𝛾t (𝑇 ′ ) + 2 ≥ 𝛾t (𝑇). Consequently, we must have equality throughout Inequality (5.10). In particular, 𝜌 o (𝑇) = 𝛾t (𝑇). Hence, we may assume that deg𝑇 (𝑣 3 ) ≥ 3. Suppose that the vertex 𝑣 3 has a descendant 𝑢 0 at distance 3 that is different from 𝑣 0 . Let 𝑢 0 𝑢 1 𝑢 2 𝑣 3 be the path from 𝑢 0 to 𝑣 3 . Using arguments similar to those used for vertices 𝑣 1 and 𝑣 2 , we may assume that deg𝑇 (𝑢 1 ) = deg𝑇 (𝑢 2 ) = 2. We now consider the tree 𝑇 ′ = 𝑇 − {𝑢 0 , 𝑢 1 , 𝑢 2 }. Every 𝛾t -set of 𝑇 ′ can be extended to a TD-set of 𝑇 by adding to it the vertices 𝑢 1 and 𝑢 2 , implying that 𝛾t (𝑇) ≤ 𝛾t (𝑇 ′ ) + 2. Let 𝑆 be a maximum open packing in 𝑇 ′ . If 𝑣 3 ∈ 𝑆, then we can replace 𝑣 3 in 𝑆 with the vertex 𝑣 1 . Hence, we can choose the set 𝑆 so that 𝑣 3 ∉ 𝑆, and so the set 𝑆 can be extended to an open packing in 𝑇 by adding to it the vertices 𝑢 0 and 𝑢 1 , implying that 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ) + 2. Applying the inductive hypothesis to 𝑇 ′ , we have 𝜌 o (𝑇 ′ ) = 𝛾t (𝑇 ′ ). Therefore, once again Inequality (5.10) holds, implying that 𝜌 o (𝑇) = 𝛾t (𝑇). Thus, we may assume that every child of 𝑣 3 different from 𝑣 2 is a leaf or a support vertex of degree 2. Suppose that the vertex 𝑣 3 has a child 𝑢 2 that is a support vertex. Let 𝑢 1 be the leaf neighbor of 𝑢 2 . We now consider the tree 𝑇 ′ = 𝑇 − {𝑣 0 , 𝑣 1 , 𝑣 2 }. Every 𝛾t -set of 𝑇 ′ can be extended to a TD-set of 𝑇 by adding to it the vertices 𝑣 1 and 𝑣 2 , implying that

122

Chapter 5. Domination in Trees

𝛾t (𝑇) ≤ 𝛾t (𝑇 ′ ) + 2. Let 𝑆 be a maximum open packing in 𝑇 ′ . If 𝑣 3 ∈ 𝑆, then we can replace 𝑣 3 in 𝑆 with the vertex 𝑢 1 . Hence, we can choose the set 𝑆 so that 𝑣 3 ∉ 𝑆, and so the set 𝑆 can be extended to an open packing in 𝑇 by adding to it the vertices 𝑣 0 and 𝑣 1 , implying that 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ) + 2. Applying the inductive hypothesis to 𝑇 ′ , we have 𝜌 o (𝑇 ′ ) = 𝛾t (𝑇 ′ ). Therefore, once again Inequality (5.10) holds, implying that 𝜌 o (𝑇) = 𝛾t (𝑇). Hence, we may assume that every child of 𝑣 3 different from 𝑣 2 is a leaf, implying that deg𝑇 (𝑣 3 ) = 3. Let 𝑢 2 be the leaf child of 𝑣 3 . We now consider the tree 𝑇 ′ = 𝑇 − 𝑣 0 . Every 𝛾t -set of 𝑇 ′ contains all of its support vertices, and therefore contains both 𝑣 2 and 𝑣 3 . Such a set can therefore be extended to a TD-set of 𝑇 by adding to it the vertex 𝑣 1 , implying that 𝛾t (𝑇) ≤ 𝛾t (𝑇 ′ ) + 1. Let 𝑆 be a maximum open packing in 𝑇 ′ . If 𝑣 2 ∈ 𝑆, then we can replace 𝑣 2 in 𝑆 with the vertex 𝑢 2 . Hence, we can choose the set 𝑆 so that 𝑣 2 ∉ 𝑆, and so the set 𝑆 can be extended to an open packing in 𝑇 by adding to it the vertex 𝑣 0 , implying that 𝜌 o (𝑇) ≥ 𝜌 o (𝑇 ′ ) + 1. Therefore, once again Inequality (5.9) holds, implying that 𝜌 o (𝑇) = 𝛾t (𝑇).

5.4

Independent Domination in Trees

In this section, we present two upper bounds on the independent domination number of a tree, and we discuss trees having unique minimum dominating sets.

5.4.1

Independent Domination Bounds in Trees

Recall that in Section 4.3.3, we presented the result due to Favaron √ [274] in 1988 that if 𝐺 is an isolate-free graph of order 𝑛, then 𝑖(𝐺) ≤ 𝑛 + 2 − 2 𝑛 and this bound is tight. In the special case of trees, this bound is achievable when 𝐺 is the path 𝑃4 . However, for trees of larger order, this general upper bound is far from best possible. Since every bipartite isolate-free graph is the union of two maximal independent sets, each of which forms a dominating set in the graph, we observe that the independent domination number of a bipartite isolate-free graph is at most one-half its order. In the special case when the bipartite graph is a tree, we have the following result. Proposition 5.43 If 𝑇 is a tree of order 𝑛 ≥ 2, then 𝑖(𝑇) ≤ 12 𝑛. The upper bound in Proposition 5.43 is tight. For example, if 𝑇 = 𝑇 ′ ◦ 𝐾1 is the corona of some tree 𝑇 ′ of order 𝑘, then 𝑇 has order 𝑛 = 2𝑘 and 𝑖(𝑇) = 𝑘 = 12 𝑛. As a further example, if 𝑇 is a double star 𝑆(𝑘, 𝑘), where 𝑘 ≥ 1, then 𝑇 has order 𝑛 = 2(𝑘 + 1) and 𝑖(𝑇) = 𝑘 + 1 = 12 𝑛. In 1992 Favaron [275] proved the following upper bound on the independent domination number of a tree and characterized the trees attaining it. This bound was first conjectured in 1980 by McFall and Nowakowski [587]. Theorem 5.44 ([275]) If 𝑇 is a tree of order 𝑛 ≥ 2 with ℓ leaves, then 𝑖(𝑇) ≤ 1 3 (𝑛 + ℓ).

Section 5.5. Equality of Domination Parameters

123

The bound in Theorem 5.44 is achieved, for example, by the path 𝑃3𝑘+1 when 𝑘 ≥ 1. Other families achieving equality include, for example, when 𝑇 = 𝑃2𝑘 ◦ 𝐾1 is the corona of a path 𝑃2𝑘 , where 𝑘 ≥ 1. In this example, 𝑇 has order 𝑛 = 4𝑘 with ℓ = 2𝑘 leaves, and satisfies 𝑖(𝑇) = 2𝑘 = 31 (𝑛 + ℓ). The full list of trees attaining this bound is provided in [275]. We remark that the proof of Theorem 5.44 provided by Favaron [275] is an inductive proof that includes the characterization of the trees attaining this bound.

5.4.2

Unique Minimum Independent Dominating Sets in Trees

We conclude this section with a characterization of trees having a unique minimum independent dominating set due to Hedetniemi [437] in 2017. For a graph 𝐺 with unique 𝑖-set 𝐼, let  X(𝐼) = 𝑣 ∈ 𝐼 : |epn(𝑣, 𝐼)| ≥ 2 ,  Y (𝐼) = 𝑣 ∈ 𝐼 : |epn(𝑣, 𝐼)| = 0 , and  Z(𝐺) = 𝑣 ∈ 𝑉 (𝐺) \ 𝐼 : |N(𝑣) ∩ 𝐼 | ≥ 2 and 𝑖(𝐺 − N[𝑣]) > 𝑖(𝐺) . Let 𝐺 1 and 𝐺 2 be graphs, each of which has a unique 𝑖-set, denoted by 𝐼1 and 𝐼2 , respectively. We now define a set of operations on graphs 𝐺 1 and 𝐺 2 . Operation 1. Let 𝐺 be the graph obtained from 𝐺 1 ∪ 𝐺 2 by adding an edge 𝑢 1 𝑢 2 , where 𝑢 𝑗 ∈ 𝑉 (𝐺 𝑗 ) \ 𝐼 𝑗 for 𝑗 ∈ [2]. Operation 2. Let 𝐺 be the graph obtained from 𝐺 1 ∪ 𝐺 2 by adding a new vertex 𝑢 and edges 𝑢𝑣 1 and 𝑢𝑣 2 , where 𝑣 𝑗 ∈ X(𝐼 𝑗 ) for 𝑗 ∈ [2]. Operation 3. Let 𝐺 be the graph obtained from 𝐺 1 ∪ 𝐺 2 by adding a new vertex 𝑢 and edges 𝑢𝑣 1 and 𝑢𝑣 2 , where 𝑣 1 ∈ X(𝐼1 ) and 𝑣 2 ∈ Y (𝐼2 ). Operation 4. Let 𝐺 be the graph obtained from 𝐺 1 ∪ 𝐺 2 by adding an edge 𝑢 1 𝑢 2 , where 𝑢 1 ∈ 𝑉 (𝐺 1 ) \ 𝐼1 is adjacent to at least two vertices in 𝐼1 , and 𝑢 2 ∈ X(𝐼2 ). Operation 5. Let 𝐺 be the graph obtained from 𝐺 1 ∪ 𝐺 2 by adding an edge 𝑢 1 𝑢 2 , where 𝑢 1 ∈ Z(𝐺 1 ) and 𝑢 2 ∈ Y (𝐼2 ). We now state the characterization of trees having a unique 𝑖-set. To simplify the statement, we say that a star of order at least 3 is a big star. Theorem 5.45 ([437]) A tree 𝑇 has an unique 𝑖-set if and only if 𝑇 can be constructed from a disjoint union of isolated vertices and big stars by a finite sequence of Operations 1 through 5.

5.5

Equality of Domination Parameters

For any two graph parameters 𝜆 and 𝜇, a graph 𝐺 is said to be a (𝜆, 𝜇)-graph if 𝜆(𝐺) = 𝜇(𝐺). The problem of characterizing graphs for which two related domination parameters are equal has received much interest. The family of (𝛾, 𝑖)-graphs are especially challenging and to date only subsets of this family have been characterized. The family of (𝛾, 𝑖)-trees was first characterized by Harary and Livingston [385] in 1986, but this characterization is rather complex. Recall from Section 5.2.4

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124

that A (𝑇) denotes the set of vertices in a tree 𝑇 that are contained in every 𝛾-set of 𝑇. Similarly, let A𝑖 (𝑇) and At (𝑇) denote the sets of vertices which are contained in every 𝑖-set and in every 𝛾t -set of 𝑇, respectively. In 2000 Cockayne et al. [184] gave a different characterization in terms of the sets A (𝑇) and A𝑖 (𝑇). In 2006 Dorfling et al. [240] used a labeling method to provide a simple constructive characterization of the (𝛾, 𝑖)-trees. This labeling method also yields the (𝛾, 𝛾t )-trees. We present the labeling methods and constructions of these families here.

5.5.1

(𝜸, 𝒊)-trees

By Theorem 5.29, the (𝛾, 𝑖)-trees are precisely the (𝜌, 𝑖)-trees. Thus, the constructive characterization given by Dorfling et al. [240] of (𝜌, 𝑖)-trees yields the (𝛾, 𝑖)-trees. The key to their characterization is a labeling of the vertices that indicates the role each vertex plays in the maximum packings and minimum independent dominating sets of a tree. To aid in the construction, define a (𝜌, 𝑖)-labeling of a tree 𝑇 to be a partition 𝑆 = {𝑆 𝐴, 𝑆 𝐵 , 𝑆𝐶 , 𝑆 𝐷 } of 𝑉 such that 𝑆 𝐴 ∪ 𝑆 𝐷 is an 𝑖-set, 𝑆𝐶 ∪ 𝑆 𝐷 is a 𝜌-set, and |𝑆 𝐴 | = |𝑆𝐶 |. The label or status of a vertex 𝑣, denoted sta(𝑣), is the letter 𝑥 ∈ {𝐴, 𝐵, 𝐶, 𝐷} such that 𝑣 ∈ 𝑆 𝑥 . A labeled graph is simply one where each vertex is labeled with either 𝐴, 𝐵, 𝐶, or 𝐷. We will need two lemmas from [240]. The following lemma can be easily proven. If 𝐺 has a (𝜌, 𝑖)-labeling, then 𝜌(𝐺) = |𝑆𝐶 ∪ 𝑆 𝐷 | = |𝑆 𝐴 ∪ 𝑆 𝐷 | = 𝑖(𝐺). Conversely, if 𝜌(𝐺) = 𝑖(𝐺), then let 𝑋 be an 𝑖-set of 𝐺, 𝑌 be a 𝜌-set of 𝐺, and create a (𝜌, 𝑖)-labeling by letting 𝑆 𝐴 = 𝑋 \ 𝑌 , 𝑆 𝐵 = 𝑉 (𝐺) \ (𝑋 ∪ 𝑌 ), 𝑆𝐶 = 𝑌 \ 𝑋, and 𝑆 𝐷 = 𝑋 ∩ 𝑌 . Lemma 5.46 ([240]) A graph is a (𝜌, 𝑖)-graph if and only if it has a (𝜌, 𝑖)-labeling. Lemma 5.47 ([240]) Consider a (𝜌, 𝑖)-labeling. If 𝑣 ∈ 𝑆 𝐴 (respectively, 𝑆𝐶 ), then 𝑣 is adjacent to exactly one vertex of 𝑆𝐶 (respectively, 𝑆 𝐴), and to no vertex of 𝑆 𝐷 . Let L be the minimum family of labeled trees that: (a) contains (𝑃1 , 𝑆1 ), where the single vertex has status 𝐷, and contains (𝑃2 , 𝑆2 ), where one vertex has status 𝐴 and the other status 𝐶; and (b) is closed under the six Operations T𝑗 , where 𝑗 ∈ [6], listed below, which extend the tree 𝑇 by attaching a tree to the vertex 𝑦 ∈ 𝑉 (𝑇), called the attacher. Operation T1 . Assume sta(𝑦) ∈ {𝐴, 𝐷}. Add a vertex 𝑥 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐵. Operation T2 . Assume sta(𝑦) ∈ {𝐴, 𝐵}. Add a path 𝑥 𝑤 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐵 and sta(𝑤) = 𝐷. Operation T3 . Assume sta(𝑦) = 𝐵. Add a path 𝑥 𝑤 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐴 and sta(𝑤) = 𝐶. Operation T4 . Assume sta(𝑦) ∈ {𝐵, 𝐶}. Add a path 𝑥 𝑤 𝑧 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐵, sta(𝑤) = 𝐴, and sta(𝑧) = 𝐶. Operation T5 . Assume sta(𝑦) = 𝐴. Add a path 𝑥 𝑤 𝑧 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐵, sta(𝑤) = 𝐶, and sta(𝑧) = 𝐴. Operation T6 . Assume sta(𝑦) = 𝐵. Add a path 𝑣 𝑢 𝑥 𝑤 𝑧 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐵, sta(𝑤) = sta(𝑣) = 𝐶, and sta(𝑧) = sta(𝑢) = 𝐴.

Section 5.5. Equality of Domination Parameters

125

These operations are illustrated in Figure 5.9.

𝐴 or 𝐷

𝐴 or 𝐵

𝐵

(a) T1

𝐵

𝐵 or 𝐶

𝐶

𝐴

𝐵

𝐷

(b) T2

(c) T3

𝐴

𝐵

𝐵

𝐴

𝐶

𝐴

𝐶

𝐶

𝐴

(d) T4

𝐶

𝐴

𝐵

𝐵

(f) T6

(e) T5

Figure 5.9 The six T𝑗 operations

Theorem 5.48 ([240]) A labeled tree is a (𝜌, 𝑖)-tree if and only if it is in L. Proof It is straightforward to check that every element of L is a (𝜌, 𝑖)-tree. The proof that every (𝜌, 𝑖)-tree 𝑇 is in L is by induction on the order of 𝑇. By Lemma 5.46, 𝑇 has an (𝜌, 𝑖)-labeling 𝑆. For the base case, consider any star 𝑇. It follows easily that there is a construction of (𝑇, 𝑆) for any (𝜌, 𝑖)-labeling 𝑆 by starting with either the 𝑃1 or the 𝑃2 and repeatedly using T1 . So fix a (𝜌, 𝑖)-tree (𝑇, 𝑆), and assume that any smaller (𝜌, 𝑖)-tree is in L. We may assume that diam(𝑇) ≥ 3, since otherwise 𝑇 is a star, and the result holds. Let 𝐼 = 𝑆 𝐴 ∪ 𝑆 𝐷 and 𝑃 = 𝑆𝐶 ∪ 𝑆 𝐷 . We proceed further with the following claim. Claim 5.48.1 Let 𝑢 be any vertex of a rooted (𝜌, 𝑖)-tree (𝑇, 𝑆) other than the root, with 𝑣 the parent of 𝑢, and let (𝑇 ′ , 𝑆 ′ ) be the labeled tree formed by the deletion of 𝑇𝑢 . Suppose that (𝑇, 𝑆) can be obtained from (𝑇 ′ , 𝑆 ′ ) by attaching 𝑇𝑢 to 𝑣 using an Operation T𝑗 . Then (𝑇, 𝑆) ∈ L except possibly if 𝑗 = 3 and 𝑣 is not dominated by 𝐼 \ {𝑢}. Proof We want to show that (𝑇 ′ , 𝑆 ′ ) is a (𝜌, 𝑖)-tree, since then, by the inductive hypothesis, (𝑇 ′ , 𝑆 ′ ) ∈ L and can be extended to (𝑇, 𝑆) by using Operation T𝑗 . For any set 𝑍 ⊆ 𝑉 (𝑇), let 𝑍 ′ = 𝑍 ∩ 𝑉 (𝑇 ′ ). For all operations, the number of vertices of 𝑇𝑢 of status 𝐴 equals the number of vertices of 𝑇𝑢 of status 𝐶, and so ′ |. Since 𝑃 is a packing, 𝑃 ′ is a packing. Since 𝐼 is independent, 𝐼 ′ is |𝑆 ′𝐴 | = |𝑆𝐶

Chapter 5. Domination in Trees

126

independent. Since 𝐼 dominates 𝑇, 𝐼 ′ will dominate 𝑇 ′ provided 𝑣 is dominated by an element of 𝐼 other than 𝑢. If 𝑗 = 3, this is assumed. If 𝑗 ≠ 3, then 𝑢 has status 𝐵 and so this is necessarily the case. We return to the proof of Theorem 5.48. Let 𝑣 0 𝑣 1 . . . 𝑣 𝑑 be a longest path in 𝑇, and so 𝑑 = diam(𝑇) ≥ 3. Root the tree 𝑇 at the vertex 𝑟 = 𝑣 𝑑 . Necessarily, 𝑣 0 and 𝑣 𝑑 are leaves of 𝑇. For ease of discussion in the current proof, we call the vertices at maximum distance from 𝑟 eccentric vertices, and so 𝑣 0 is an eccentric vertex. Suppose sta(𝑣 0 ) = 𝐵. Then since 𝑣 0 is dominated by 𝐼, the vertex 𝑣 1 has status 𝐴 or 𝐷. And so (𝑇, 𝑆) ∈ L by Claim 5.48.1 with 𝑢 = 𝑣 0 and 𝑗 = 1. So we may assume that no eccentric vertex has status 𝐵. Suppose sta(𝑣 0 ) = 𝐷. Then by Lemma 5.47, sta(𝑣 1 ) = 𝐵. Since 𝑃 is a packing, any neighbor of 𝑣 1 has status 𝐴 or 𝐵. This means that 𝑣 1 has no other leaf neighbor (since a vertex with status 𝐴 has a neighbor with status 𝐶) and so has degree 2. Hence, (𝑇, 𝑆) ∈ L by Claim 5.48.1 with 𝑢 = 𝑣 1 and 𝑗 = 2. Thus, we may assume that every eccentric vertex has status 𝐴 or 𝐶. So, by Lemma 5.47, every vertex at distance 2 from an eccentric vertex has status 𝐵. In particular, this means that 𝑣 1 has degree 2. Suppose sta(𝑣 0 ) = 𝐶. Then, (𝑇, 𝑆) ∈ L by Claim 5.48.1 with 𝑢 = 𝑣 1 and 𝑗 = 3, unless 𝑣 2 has no neighbor in 𝐼 \ {𝑢}. So, suppose that is the case. Then sta(𝑣 3 ) ∈ {𝐵, 𝐶}. If 𝑣 2 has degree 2, then (𝑇, 𝑆) ∈ L by Claim 5.48.1 with 𝑢 = 𝑣 2 and 𝑗 = 4. Hence, assume deg(𝑣 2 ) ≥ 3. This means that 𝑣 2 has a neighbor 𝑢 1 ≠ 𝑣 3 that has status 𝐵 or 𝐶. Since 𝐼 dominates 𝑢 1 , the vertex 𝑢 1 has a neighbor 𝑢 0 with sta(𝑢 0 ) ∈ { 𝐴, 𝐷}. Note that 𝑢 0 is an eccentric vertex, so (as above) deg(𝑢 1 ) = 2. By Lemma 5.47 and the above assumptions, sta(𝑢 0 ) = 𝐴 and sta(𝑢 1 ) = 𝐶. But 𝑣 2 can only have one neighbor with status 𝐶, and so has degree 3. Thus, (𝑇, 𝑆) ∈ L by Claim 5.48.1 with 𝑢 = 𝑣 2 and 𝑗 = 6. Hence, we may assume that all eccentric vertices have status 𝐴. This means that all neighbors of 𝑣 2 , apart from 𝑣 3 , have status 𝐶, and so 𝑣 2 has degree 2. It follows that sta(𝑣 3 ) = 𝐴. Thus, (𝑇, 𝑆) ∈ L by Claim 5.48.1 with 𝑢 = 𝑣 2 and 𝑗 = 5. The next corollary follows directly from Theorem 5.42, Lemma 5.46, and Theorem 5.48. Corollary 5.49 ([240]) The (𝛾, 𝑖)-trees are precisely those trees 𝑇 such that (𝑇, 𝑆) ∈ L for some labeling 𝑆.

5.5.2

(𝜸, 𝜸t )-trees

As before by Theorem 5.29, the (𝛾, 𝛾t )-trees are precisely the (𝜌, 𝛾t )-trees. And again the key to the constructive characterization of (𝜌, 𝛾t )-trees is to find a labeling of the vertices that indicates the role each vertex plays in the sets associated with both parameters. We adopt the same terminology used for (𝜌, 𝑖)-trees. In particular, a (𝜌, 𝛾t )labeling of a graph 𝐺 = (𝑉, 𝐸) as a partition 𝑆 = (𝑆 𝐴, 𝑆 𝐵 , 𝑆𝐶 , 𝑆 𝐷 ) of 𝑉 such that 𝑆 𝐴 ∪ 𝑆 𝐷 is a 𝛾t -set, 𝑆𝐶 ∪ 𝑆 𝐷 is a 𝜌-set, and |𝑆 𝐴 | = |𝑆𝐶 |.

Section 5.5. Equality of Domination Parameters

127

Lemma 5.50 ([240]) A graph is a (𝜌, 𝛾t )-graph if and only if it has a (𝜌, 𝛾t )labeling.

The smallest (𝛾, 𝛾t )-tree is the path 𝑃4 . It has a unique labeling as a (𝜌, 𝛾t )-tree, where the leaves have status 𝐶 and the internal vertices have status 𝐴. Let Lt be the minimum family of labeled trees that: (a) contains a labeled 𝑃4 , where the leaves have status 𝐶 and the internal vertices have status 𝐴, and (b) is closed under Operations U𝑖 for 𝑖 ∈ [5], listed below, which extend the tree 𝑇 by attaching a tree to the vertex 𝑦 ∈ 𝑉 (𝑇), called the attacher. Operation U1 . Assume sta(𝑦) ∈ {𝐴, 𝐷}. Add a vertex 𝑥 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐵. Operation U2 . Assume sta(𝑦) = 𝐴. Add a path 𝑥 𝑤 and the edge 𝑥𝑦. Let sta(𝑥) = 𝐴 and sta(𝑤) = 𝐶. Operation U3 . Take a vertex 𝑦 of status 𝐵 which has no neighbor of status 𝐶, add a labeled 𝑃4 , and join 𝑦 to a leaf of the 𝑃4 . Operation U4 . Add a labeled 𝑃4 , and join a vertex 𝑦 of status 𝐵 to an internal vertex of the 𝑃4 . Operation U5 . Add a labeled 𝑃4 and a vertex 𝑦 ′ labeled 𝐵, and attach to a vertex 𝑦 of status 𝐵 or 𝐶 the added vertex 𝑦 ′ and join 𝑦 ′ to an internal vertex of the added labeled 𝑃4 . These five operations are illustrated in Figure 5.10.

𝐴 or 𝐷

𝐴

𝐵

(a) U1

𝐵 𝑦

𝐶

𝐶

𝐴

(b) U2

𝐴

𝐴

𝐶

𝐵

𝐴

𝐴 𝐶

(c) U3 ; 𝑦 has no prior 𝑆𝐶 neighbor

𝐵 or 𝐶

(d) U4

𝐵

𝐴

𝐴

𝐶

𝐶 (e) U5

Figure 5.10 The five U𝑖 operations

𝐶

128

Chapter 5. Domination in Trees

Theorem 5.51 ([240]) A labeled tree is a (𝜌, 𝛾t )-tree if and only if it can be obtained from a labeled 𝑃4 using the Operations U𝑖 for 𝑖 ∈ [5]. The next corollary follows directly from Theorem 5.42, Lemma 5.50, and Theorem 5.48. Corollary 5.52 ([240]) The (𝛾, 𝛾t )-trees are precisely those trees 𝑇 such that (𝑇, 𝑆) ∈ Lt for some labeling 𝑆.

5.5.3 Summary We conclude this section by mentioning a couple of related concepts. As we have seen, much interest has been shown in characterizing graphs achieving equality in an inequality between two parameters. For example, the (𝛾, 𝑖)-graphs are graphs achieving equality in the relationship 𝛾(𝐺) ≤ 𝑖(𝐺). Note that if equality is reached between 𝛾(𝐺) and 𝑖(𝐺), then the graph 𝐺 has a 𝛾-set that is also an 𝑖-set. Haynes and Slater [436] considered graphs 𝐺 for which every 𝛾-set of 𝐺 is also an 𝑖-set of 𝐺, and hence introduced the concept of strong equality between parameters related by an inequality. In particular, 𝛾(𝐺) is said to be strongly equal to 𝑖(𝐺) if every 𝛾-set is also an 𝑖-set of 𝐺. Haynes et al. [428] gave a constructive characterization of the trees 𝑇 with strong equality between 𝛾(𝑇) and 𝑖(𝑇). A graph 𝐺 is said to be domination perfect if 𝛾(𝐺 ′ ) = 𝑖(𝐺 ′ ) for all subgraphs 𝐺 ′ of 𝐺. Domination perfect trees are characterized as follows: A tree is domination perfect if and only if it does not contain two adjacent vertices of degree 3 or more. (This characterization follows as a corollary of the results from any of [315, 693, 790].)

Chapter 6

Upper Bounds in Terms of Minimum Degree 6.1 Introduction As previously mentioned, since the decision problems related to the domination number, the total domination number, and the independent domination number are all NP-complete, it is of interest to determine good upper bounds on these parameters. In Chapter 4 we presented some of the more basic bounds. In this chapter we continue with bounds on these parameters in terms of the order of the graph and its minimum degree.

6.2

Bounds on the Domination Number

In this section, we present the classical bound 𝛾(𝐺) ≤ 52 𝑛 of McCuaig and Shepherd on the domination number of a connected graph 𝐺 of order 𝑛 ≥ 8 and minimum degree 𝛿(𝐺) ≥ 2 using the proof technique of what they coined 25 -minimal graphs. Using vertex-disjoint path covers, we present the important bound 𝛾(𝐺) ≤ 38 𝑛 of Reed on the domination number of a graph 𝐺 of order 𝑛 and minimum degree 𝛿(𝐺) ≥ 3. For larger minimum degrees, we present the impressive results of Bujtás using weighting arguments and discharging methods. We illustrate her proof techniques 4 to obtain the bounds 𝛾(𝐺) ≤ 11 𝑛 and 𝛾(𝐺) ≤ 13 𝑛 when the graph 𝐺 has minimum degree 𝛿(𝐺) ≥ 4 and 𝛿(𝐺) ≥ 5, respectively.

6.2.1

Minimum Degree One

As given in Theorem 4.21, the domination number of any isolate-free graph is at most half its order. A characterization of graphs obtaining this bound is presented in Theorem 4.24. For completeness, we restate these results from Chapter 4. Lemma 6.1 ([84]) Every isolate-free graph 𝐺 contains a 𝛾-set 𝐷 such that epn[𝑣, 𝐷] ≠ ∅ for every vertex 𝑣 ∈ 𝐷. © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_6

129

Chapter 6. Upper Bounds in Terms of Minimum Degree

130

Theorem 6.2 ([622]) If 𝐺 is an isolate-free graph of order 𝑛, then 𝛾(𝐺) ≤ 21 𝑛. Theorem 6.3 ([633]) If 𝐺 is an isolate-free graph of even order 𝑛, then 𝛾(𝐺) = 12 𝑛 if and only if every component of 𝐺 is a 4-cycle or 𝐺 = 𝐻 ◦ 𝐾1 for some graph 𝐻.

6.2.2

Minimum Degree Two

In 1973 Blank [79] proved that the 12 -bound on the domination number of an isolatefree, connected graph 𝐺 can be improved to a 25 -bound if we restrict the minimum degree to 𝛿(𝐺) ≥ 2 and the order to 𝑛 ≥ 8. Theorem 6.4 ([79]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then   𝑛+2 2 𝛾(𝐺) ≤ max , 𝑛 . 3 5 As an immediate consequence of Theorem 6.4, we have the following result. Theorem 6.5 ([79]) If 𝐺 is a connected graph of order 𝑛 ≥ 8 with 𝛿(𝐺) ≥ 2, then 𝛾(𝐺) ≤ 25 𝑛. We remark that this important 1973 paper by Blank [79] was written in Russian. Furthermore, Blank used the notation “external stability number” for the “domination number.” Hence, the paper was not easily traceable by researchers in domination theory at the time. In 1989 McCuaig and Shepherd [586] reproved Theorem 6.4. Their proof provides insight into the structure of graphs achieving the 25 -upper bound. Their result also gives a characterization of the infinite family of graphs achieving equality in the upper bound when the order 𝑛 ≥ 15. In order to present the McCuaig-Shepherd proof, the following definition is needed. Definition 6.6 ([586]) A graph 𝐺 of order 𝑛 ≥ 3 is a 25 -minimal graph if 𝐺 is edge minimal with respect to satisfying all three of the following conditions (and so, deleting any edge means that at least one of the following three conditions is no longer satisfied): (a) 𝛿(𝐺) ≥ 2. (b) 𝐺 is connected. (c) 𝛾(𝐺) ≥ 25 𝑛. Let Bdom = {𝐵1 , 𝐵2 , . . . , 𝐵7 } be the family of seven graphs (one of order four and six of order seven) shown in Figure 6.1. Each graph 𝐺 ∈ Bdom of order 𝑛 satisfies 𝛿(𝐺) = 2 and 𝛾(𝐺) > 25 𝑛. The following properties of graphs in the family Bdom will be required. Proposition 6.7 ([79]) If 𝐺 ∈ Bdom has order 𝑛, then the following hold: (a) 𝛾(𝐺) = 13 (𝑛 + 2). (b) Every vertex belongs to some 𝛾-set of 𝐺. (c) For every vertex 𝑣, 𝛾(𝐺 − 𝑣) = 𝛾(𝐺) − 1.

Section 6.2. Bounds on the Domination Number

(a) 𝐵1

(d) 𝐵4

131

(b) 𝐵2

(e) 𝐵5

(c) 𝐵3

(f) 𝐵6

(g) 𝐵7

Figure 6.1 The family Bdom of seven graphs 𝐺 satisfying 𝛾(𝐺) > 25 𝑛

McCuaig and Shepherd [586] constructed the following (infinite) family of graphs. We define a 5-key to be a graph of order 5 obtained from a 4-cycle by adding a new vertex and joining this vertex to exactly one vertex of the cycle. We define a unit to be a graph that is isomorphic to a 5-cycle or to a 5-key. We call a unit a cycle unit or a key unit if it is a 5-cycle or a 5-key, respectively. In each cycle unit, we choose two nonadjacent vertices in the unit and designate them as the link vertices of the unit, and in each key unit we designate the vertex of degree 1 as the link vertex of the unit. Let Fdom be the family of all graphs 𝐺 such that either 𝐺 = 𝐶5 or 𝐺 can be obtained from the disjoint union of at least two units, by adding edges between link vertices so that the resulting graph is connected and each added edge is a bridge of 𝐺. A graph 𝐺 in the family Fdom with two cycle units and two key units is shown in Figure 6.2, where the link vertices are highlighted. 2 5 -minimal

Figure 6.2 A graph 𝐺 in the family Fdom Let Fsmall = {𝐹1 , 𝐹2 , . . . , 𝐹6 } be the family of six graphs shown in Figure 6.3. We shall need the following properties of graphs in the family Fdom ∪ Fsmall . Lemma 6.8 ([79]) If 𝐺 ∈ Fdom ∪ Fsmall has order 𝑛, then the following hold: (a) 𝛾(𝐺) = 25 𝑛 and 𝐺 is a 25 -minimal graph. (b) Every vertex belongs to some 𝛾-set of 𝐺. (c) If 𝐺 ∈ Fsmall , then for every vertex 𝑣, 𝛾(𝐺 − 𝑣) = 𝛾(𝐺) − 1, except when 𝐺 = 𝐹1 and 𝑣 is a vertex of degree 2.

Chapter 6. Upper Bounds in Terms of Minimum Degree

132

(a) 𝐹1

(b) 𝐹2

(d) 𝐹4

(c) 𝐹3

(e) 𝐹5

(f) 𝐹6

Figure 6.3 The family Fsmall of six graphs

Proof (a) Suppose firstly that 𝐺 ∈ Fsmall has order 𝑛. If 𝐺 = 𝐹1 , then 𝑛 = 5 and 𝛾(𝐺) = 2 = 25 𝑛, while if 𝐺 ∈ Fsmall \ {𝐹1 }, then 𝑛 = 10 and 𝛾(𝐺) = 4 = 25 𝑛. Suppose next that 𝐺 ∈ Fdom has order 𝑛. Let 𝐺 have 𝑘 ≥ 2 units, and so 𝑛 = 5𝑘. Every dominating set in 𝐺 contains at least two vertices from each unit of 𝐺, implying that 𝛾(𝐺) ≥ 2𝑘. The set of all link vertices, together with a vertex from each key unit that is a distance 3 from its link vertex, forms a dominating set of 𝐺. Hence, there is a dominating set in 𝐺 that contains exactly two vertices from each unit of 𝐺, and so 𝛾(𝐺) ≤ 2𝑘. Consequently, 𝛾(𝐺) = 2𝑘 = 25 𝑛. Further, if 𝐺 ∈ Fdom ∪ Fsmall and 𝑒 is an arbitrary edge of 𝐺, then the removal of 𝑒 produces a disconnected graph or creates a vertex of degree 1. Thus, 𝐺 is a 25 -minimal graph. This completes the proof of (a). The proofs of (b) and (c) follow readily from the structure of graphs in the family Fdom ∪ Fsmall . We proceed further with a series of preliminary results that will be helpful when we present a characterization of 25 -minimal graphs. Recall that for paths 𝑃𝑛 and   cycles 𝐶𝑛 for 𝑛 ≥ 3, 𝛾(𝑃𝑛 ) = 𝛾(𝐶𝑛 ) = 𝑛3 . This readily yields the following result. Proposition 6.9 A cycle 𝐶𝑛 is a 25 -minimal graph if and only if 𝑛 ∈ {4, 5, 7, 10}. A daisy with 𝑘 ≥ 2 petals is a connected graph that can be constructed from 𝑘 ≥ 2 disjoint cycles by identifying a set of 𝑘 vertices, one from each cycle, into one vertex. In particular, if the 𝑘 cycles have lengths 𝑛1 , 𝑛2 , . . . , 𝑛 𝑘 , we denote the daisy by 𝐷 (𝑛1 , 𝑛2 , . . . , 𝑛 𝑘 ). Further, if 𝑛 = 𝑛1 = 𝑛2 = · · · = 𝑛 𝑘 , then we write 𝐷 (𝑛1 , 𝑛2 , . . . , 𝑛 𝑘 ) simply as 𝐷 𝑘 (𝑛). For example, the graph 𝐵2 in Figure 6.1(b) is a daisy 𝐷 2 (4) = 𝐷 (4, 4). The graphs 𝐹2 and 𝐹4 in Figure 6.3(b) and (d) are the daisies 𝐷 (4, 7) and 𝐷 3 (4) = 𝐷 (4, 4, 4), respectively. A simple proof by induction on the number of petals of a daisy yields the following upper bound on its domination number in terms of its order. Proposition 6.10 If 𝐺 is a daisy 𝐷 (𝑛1 , 𝑛2 , . . . , 𝑛 𝑘 ) of order 𝑛 with 𝑘 ≥ 2 petals, then 𝛾(𝐺) ≤ 13 (𝑛 + 2), with equality if and only if 𝑛𝑖 ≡ 1 (mod 3) for all 𝑖 ∈ [𝑘].

Section 6.2. Bounds on the Domination Number

133

By Proposition 6.10, we can readily determine the 52 -minimal daisies.  Proposition 6.11 A daisy 𝐺 is a 25 -minimal graph if and only if 𝐺 ∈ 𝐷 2 (4), 𝐷 3 (4), 𝐷 (4, 7) . We shall need the following lemma about subdividing certain edges in a graph. Lemma 6.12 If 𝐺 is the graph obtained from a graph 𝐺 ′ by subdividing an edge three times, then 𝛾(𝐺) = 𝛾(𝐺 ′ ) + 1. Proof Let 𝐺 be obtained from a graph 𝐺 ′ by subdividing an edge 𝑒 = 𝑢𝑣 three times. Let 𝑣 1 , 𝑣 2 , and 𝑣 3 be the three new vertices, where 𝑢 𝑣 1 𝑣 2 𝑣 3 𝑣 is a path in 𝐺. Let 𝑆 ′ be a 𝛾-set of 𝐺 ′ . If {𝑢, 𝑣} ⊆ 𝑆 ′ , let 𝑆 = 𝑆 ′ ∪ {𝑣 2 }. If 𝑢 ∈ 𝑆 ′ and 𝑣 ∉ 𝑆 ′ , let 𝑆 = 𝑆 ′ ∪ {𝑣 3 }. If 𝑣 ∈ 𝑆 ′ and 𝑢 ∉ 𝑆 ′ , let 𝑆 = 𝑆 ′ ∪ {𝑣 1 }. If 𝑢 ∉ 𝑆 ′ and 𝑣 ∉ 𝑆 ′ , let 𝑆 = 𝑆 ′ ∪ {𝑣 2 }. In each case, the resulting set 𝑆 is a dominating set of 𝐺, implying that 𝛾(𝐺) ≤ |𝑆| = |𝑆 ′ | + 1 = 𝛾(𝐺 ′ ) + 1. Conversely, among all 𝛾-sets of 𝐺, let 𝑆 be chosen to contain as few vertices from the set {𝑣 1 , 𝑣 2 , 𝑣 3 } as possible. Suppose that 𝑣 1 ∈ 𝑆. If 𝑣 2 or 𝑣 3 belongs to 𝑆, then we can replace it with the vertex 𝑣, contradicting our choice of 𝑆. Hence, neither 𝑣 2 nor 𝑣 3 belongs to 𝑆, implying that 𝑣 ∈ 𝑆 in order to dominate 𝑣 3 . In this case, we let 𝑆 ′ = 𝑆 \ {𝑣 1 }. Analogously, if 𝑣 3 ∈ 𝑆, then neither 𝑣 1 nor 𝑣 2 belongs to 𝑆, implying that 𝑢 ∈ 𝑆. In this case, we let 𝑆 ′ = 𝑆 \ {𝑣 3 }. If 𝑣 2 ∈ 𝑆, then neither 𝑣 1 nor 𝑣 3 belongs to 𝑆, and in this case we let 𝑆 ′ = 𝑆 \ {𝑣 2 }. In each case, the resulting set 𝑆 ′ is a dominating set of 𝐺 ′ , implying that 𝛾(𝐺 ′ ) ≤ |𝑆 ′ | = |𝑆| − 1 = 𝛾(𝐺) − 1. Consequently, 𝛾(𝐺) = 𝛾(𝐺 ′ ) + 1. For 𝑛1 , 𝑛2 ≥ 3 and 𝑘 ≥ 1, a dumbbell 𝐷 (𝑛1 , 𝑛2 , 𝑘) is the graph obtained from two disjoint cycles 𝐶𝑛1 and 𝐶𝑛2 by adding an edge joining the two cycles and subdividing this edge 𝑘 − 1 times, resulting in vertices joined (connected) by a path of length 𝑘. The dumbbells 𝐷 (5, 5, 1), 𝐷 (4, 4, 3), and 𝐷 (4, 5, 2) are shown in Figure 6.4.

(a) 𝐷 (5, 5, 1)

(b) 𝐷 (4, 4, 3)

(c) 𝐷 (4, 5, 2)

Figure 6.4 The dumbbells 𝐷 (5, 5, 1), 𝐷 (4, 4, 3), and 𝐷 (4, 5, 2)   Using the fact that 𝛾(𝑃𝑛 ) = 𝛾(𝐶𝑛 ) = 𝑛3 for 𝑛 ≥ 3, it is a relatively simple exercise to determine the 25 -minimal dumbbells.  Proposition 6.13 A dumbbell 𝐺 is a 25 -minimal graph if and only if 𝐺 ∈ 𝐷 (5, 5, 1), 𝐷 (4, 4, 3), 𝐷 (4, 5, 2) ⊂ Fdom . Recall that the removal of an edge from a graph 𝐺 cannot decrease the domination number; that is, if 𝑒 is an edge of a graph 𝐺, then 𝛾(𝐺 − 𝑒) ≥ 𝛾(𝐺). In particular,

134

Chapter 6. Upper Bounds in Terms of Minimum Degree

if 𝐺 is a 52 -minimal graph, then the removal of an edge results in a graph that no longer satisfies condition (a) or condition (b) (or both conditions (a) and (b)) of Definition 6.6. We state this observation formally as follows. Observation 6.14 If 𝐺 is a 25 -minimal graph, then every edge of 𝐺 is a bridge of 𝐺 or is incident with a vertex of degree 2 in 𝐺, that is, if 𝑒 ∈ 𝐸 (𝐺), then 𝑒 is a bridge of 𝐺 or 𝛿(𝐺 − 𝑒) = 1. We next consider 25 -minimal graphs of order 𝑛 ≤ 7. Lemma 6.15 ([79]) Let 𝐺 be a connected graph of order 𝑛 ≤ 7 with 𝛿(𝐺) ≥ 2. Then, 𝐺 is a 25 -minimal graph if and only if 𝐺 ∈ {𝐵1 , 𝐵2 , 𝐵4 } ⊂ Bdom or 𝐺 = 𝐶5 ∈ Fdom or 𝐺 = 𝐹1 ∈ Fsmall . Proof If 𝐺 ∈ {𝐵1 , 𝐵2 , 𝐵4 , 𝐶5 , 𝐹1 }, then it is straightforward to check that 𝐺 is a 2 2 5 -minimal graph. To prove the necessity, suppose that 𝐺 is a 5 -minimal graph of 2 order 𝑛 ≤ 7. If 𝑛 = 3, then 𝐺 = 𝐾3 and 𝛾(𝐺) = 1 < 5 𝑛, a contradiction. If 𝑛 = 4, then since 𝛿(𝐺) ≥ 2, 𝐺 is either 𝐶4 , 𝐾4 − 𝑒 or 𝐾4 , but 𝛾(𝐾4 − 𝑒) = 𝛾(𝐾4 ) = 1. Thus, 𝐺 = 𝐶4 = 𝐵1 . If 𝑛 = 5, then 𝐺 is one of eleven possible connected graphs having minimum degree 2, but seven of these have 𝛾(𝐺) = 1. Of the remaining four, two are not edge-minimal with respect to the three properties, leaving either 𝐺 = 𝐶5 or 𝐺 = 𝐹1 . Suppose that 𝑛 = 6. If Δ(𝐺) ≥ 4, then it is immediate that 𝛾(𝐺) ≤ 2. If Δ(𝐺) = 2, then 𝐺 = 𝐶6 and 𝛾(𝐺) = 2. If Δ(𝐺) = 3, then let 𝑣 be a vertex of maximum degree 3 in 𝐺, and let 𝑢 and 𝑤 be the two vertices not adjacent to 𝑣. If 𝑢𝑤 is an edge, then 𝛾(𝐺) = 2. If 𝑢𝑤 is not an edge, then 𝑢 and 𝑤 have a common neighbor, which together with the vertex 𝑣 forms a dominating set of 𝐺, and so 𝛾(𝐺) = 2. Thus, in all cases, if 𝑛 = 6, then 𝛾(𝐺) ≤ 2 < 25 𝑛, a contradiction. If 𝑛 = 7 and Δ(𝐺) = 2, then 𝐺 = 𝐶7 = 𝐵4 . Hence, we may assume that 𝑛 = 7 and Δ(𝐺) ≥ 3, for otherwise the desired result follows. If Δ(𝐺) ≥ 5, then 𝛾(𝐺) = 2 < 25 𝑛, a contradiction. Hence, Δ(𝐺) = 3 or Δ(𝐺) = 4. Suppose that Δ(𝐺) = 4. Let 𝑣 be a vertex of maximum degree 4 and let N(𝑣) = {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 }. Let 𝐴 = 𝑉 \ N[𝑣] = {𝑣 5 , 𝑣 6 }. If one vertex dominates the two vertices in 𝐴, then 𝛾(𝐺) = 2, a contradiction. Hence, two vertices are needed to dominate 𝐴. Renaming vertices if necessary, this implies that N(𝑣 5 ) = {𝑣 1 , 𝑣 2 } and N(𝑣 6 ) = {𝑣 3 , 𝑣 4 }. By the edge minimality of 𝐺, the graph 𝐺 is determined. Hence, 𝐺 = 𝐵2 . Suppose that Δ(𝐺) = 3. Among all vertices of maximum degree 3 in 𝐺, let 𝑣 be chosen to have the minimum number of edges in the induced subgraph 𝐺 [N(𝑣)]. Let N(𝑣) = {𝑣 1 , 𝑣 2 , 𝑣 3 } and let 𝐵 = 𝑉 \ N[𝑣] = {𝑣 4 , 𝑣 5 , 𝑣 6 }. If one vertex dominates the set 𝐵, then 𝛾(𝐺) = 2, a contradiction. Hence, two vertices are needed to dominate 𝐵. In particular, every vertex in N(𝑣) dominates at most two vertices in 𝐵. Further, 𝐺 [𝐵] contains at most one edge. Since 𝛿(𝐺) ≥ 2, these observations imply that there are at least four edges with exactly one end in 𝐵, and therefore, by the Pigeonhole Principle, there is a vertex in N(𝑣) adjacent to exactly two vertices of 𝐵. Renaming vertices if necessary, we may assume that 𝑣 1 is adjacent to exactly two vertices of 𝐵, say 𝑣 4 and 𝑣 5 . If 𝑣 6 is adjacent to both 𝑣 2 and 𝑣 3 , then {𝑣 1 , 𝑣 6 }

Section 6.2. Bounds on the Domination Number

135

is a dominating set of 𝐺, and so 𝛾(𝐺) = 2, a contradiction. Thus, 𝑣 6 is adjacent to exactly one of 𝑣 2 and 𝑣 3 , say 𝑣 3 , and to exactly one of 𝑣 4 and 𝑣 5 , say 𝑣 5 . This in turn implies that there is no edge in the subgraph induced by the open neighborhood of 𝑣 1 . Hence, by our choice of the vertex 𝑣, there is no edge in 𝐺 [N(𝑣)]. The vertex 𝑣 2 is therefore adjacent to at least one of 𝑣 4 and 𝑣 5 . If 𝑣 2 is not adjacent to 𝑣 4 , then both 𝑣 2 𝑣 5 and 𝑣 3 𝑣 4 are edges. However, in this case, {𝑣 3 , 𝑣 5 } is a dominating set of 𝐺, and so 𝛾(𝐺) = 2, a contradiction. Hence, 𝑣 2 𝑣 4 is an edge. However, 𝐺 now contains a 7-cycle, namely 𝑣 𝑣 3 𝑣 6 𝑣 5 𝑣 1 𝑣 4 𝑣 2 𝑣, as a proper subgraph, contradicting the fact that 𝐺 is a 52 -minimal graph. In what follows in the remaining part of this Section 6.2.2, if 𝐺 is a graph with minimum degree at least 2, then we let L be the set of all vertices of degree at least 3 in 𝐺, and let S = 𝑉 \ L, that is,   L = 𝑣 ∈ 𝑉 : deg𝐺 (𝑣) ≥ 3 and S = 𝑣 ∈ 𝑉 : deg𝐺 (𝑣) = 2 . We call a vertex in L a large vertex, and a vertex in S a small vertex. For 𝑘 ≥ 3, we define a 𝑘-handle to be a 𝑘-cycle that contains exactly one large vertex. For 𝑘 ≥ 1, a 𝑘-linkage is a path on 𝑘 + 2 vertices that starts and ends at distinct large vertices and with 𝑘 internal vertices of degree 2 in 𝐺. A handle is a 𝑘-handle for some 𝑘 ≥ 3, and a linkage is a 𝑘-linkage for some 𝑘 ≥ 1. We are now in a position to present a characterization of the 25 -minimal graphs. Theorem 6.16 ([586]) Let 𝐺 be a connected graph of order 𝑛 ≥ 3 with 𝛿(𝐺) ≥ 2. Then, 𝐺 is a 25 -minimal graph if and only if 𝐺 ∈ {𝐵1 , 𝐵2 , 𝐵4 } ⊂ Bdom or 𝐺 ∈ Fdom ∪ Fsmall . Proof The sufficiency follows from Lemmas 6.8(a) and 6.15. To prove the necessity, suppose to the contrary, that the theorem is false. Among all counterexamples, let 𝐺 be chosen to be a 25 -minimal graph of minimum order 𝑛. Thus, 𝐺 ∉ {𝐵1 , 𝐵2 , 𝐵4 } and 𝐺 ∉ Fdom ∪ Fsmall . By Lemma 6.15, the result is true for 𝑛 ≤ 7. Hence, 𝑛 ≥ 8. By our choice of 𝐺, the result is true for all 25 -minimal graphs 𝐺 ′ of order 𝑛′ , where 𝑛′ < 𝑛. If 𝐺 = 𝐶𝑛 , then by Proposition 6.9, 𝐺 = 𝐶10 = 𝐹6 ∈ Fsmall , a contradiction. Hence, 𝐺 is not a cycle. Thus, Δ(𝐺) ≥ 3, and so |L| ≥ 1. If |L| = 1, then 𝐺 is a daisy and, by Proposition 6.11, we have 𝐺 = 𝐵2 or 𝐺 ∈ {𝐹2 , 𝐹4 } ⊂ Fsmall , a contradiction. If |L| = 2, then 𝐺 is a dumbbell and, by Proposition 6.13, we have 𝐺 ∈ {𝐷 (5, 5, 1), 𝐷 (4, 4, 3), 𝐷 (4, 5, 2)} ⊂ Fdom , a contradiction. Hence, |L| ≥ 3. We proceed further with the following structural properties of the graph 𝐺. Claim 6.16.1 The set L is an independent set. Proof Suppose, to the contrary, that L is not an independent set. Let 𝑒 = 𝑣 1 𝑣 2 be an edge of 𝐺, where 𝑣 1 , 𝑣 2 ∈ L. By Observation 6.14, the edge 𝑒 is a bridge of 𝐺. Let 𝐺 1 = (𝑉1 , 𝐸 1 ) and 𝐺 2 = (𝑉2 , 𝐸 2 ) be the two components of 𝐺 − 𝑒, where 𝑣 𝑖 ∈ 𝑉𝑖 for 𝑖 ∈ [2]. For 𝑖 ∈ [2], let |𝑉𝑖 | = 𝑛𝑖 , and so 𝑛 = 𝑛1 + 𝑛2 . We note that 𝛾(𝐺) ≤ 𝛾(𝐺 1 ) + 𝛾(𝐺 2 ). Further, for 𝑖 ∈ [2], the graph 𝐺 𝑖 is edge minimal with respect to the two conditions: (i) 𝐺 𝑖 is connected and (ii) 𝛿(𝐺 𝑖 ) ≥ 2. If 𝐺 𝑖 is not a 25 -minimal graph for some 𝑖 ∈ [2], then 𝛾(𝐺 𝑖 ) < 25 𝑛𝑖 . If neither 𝐺 1 nor 𝐺 2 is a

136 2 5 -minimal

Chapter 6. Upper Bounds in Terms of Minimum Degree

graph, then 𝛾(𝐺) ≤ 𝛾(𝐺 1 ) + 𝛾(𝐺 2 ) < 25 𝑛1 + 25 𝑛2 = 25 𝑛, a contradiction. Hence, renaming 𝐺 1 and 𝐺 2 if necessary, we may assume that 𝐺 1 is a 25 -minimal graph. Since 𝑛1 < 𝑛, the graph 𝐺 1 cannot be a counterexample to the theorem by our choice of 𝐺. Thus, 𝐺 1 ∈ {𝐵1 , 𝐵2 , 𝐵4 } ⊂ Bdom or 𝐺 ∈ Fdom ∪ Fsmall . Suppose that 𝐺 1 ∈ Bdom . If 𝛾(𝐺 2 ) < 25 𝑛2 , then 𝛾(𝐺) < 25 𝑛, a contradiction. Hence, 𝛾(𝐺 2 ) ≥ 25 𝑛2 , implying that 𝐺 2 ∈ Bdom or 𝐺 2 ∈ Fdom ∪ Fsmall . By Proposition 6.7(b) and Lemma 6.8(b), every vertex 𝑣 in a graph in Bdom , Fdom , or Fsmall belongs to some 𝛾-set of 𝐺. Thus, there exists a 𝛾-set, 𝐷 2 say, of 𝐺 2 that contains the vertex 𝑣 2 . The set 𝐷 2 can be extended to a dominating set of 𝐺 by adding a 𝛾-set of 𝐺 1 − 𝑣 1 to it. Thus,
by Proposition 6.7, we have 𝛾(𝐺) ≤ |𝐷 2 | + 𝛾(𝐺 1 − 𝑣 1 ) = 𝛾(𝐺 2 ) + 𝛾(𝐺 1 ) − 1 = 𝛾(𝐺 2 ) + 13 (𝑛1 + 2) − 1 = 𝛾(𝐺 2 ) + 13 (𝑛1 − 1). If 𝐺 2 ∈ Bdom , then 𝛾(𝐺 2 ) = 13 (𝑛2 +2), implying that 𝛾(𝐺) ≤ 13 (𝑛2 +2) + 13 (𝑛1 −1) = 13 (𝑛 +1) < 25 𝑛, noting that 𝑛 ≥ 8. If 𝐺 2 ∈ Fdom ∪ Fsmall , then 𝛾(𝐺 2 ) = 25 𝑛2 , implying that 𝛾(𝐺) ≤ 2 1 2 2 2 5 𝑛2 + 3 (𝑛1 − 1) < 5 𝑛1 + 5 𝑛2 = 5 𝑛. In both cases we produce a contradiction. Hence, 𝐺 1 ∉ Bdom , and so 𝐺 1 ∈ Fdom ∪ Fsmall . Thus, 𝛾(𝐺 1 ) = 25 𝑛1 . By Lemma 6.8(b), there exists a 𝛾-set, 𝐷 1 say, of 𝐺 1 that contains the vertex 𝑣 1 . If 𝐺 2 is not a 25 -minimal graph, then 𝛾(𝐺) ≤ 𝛾(𝐺 1 ) + 𝛾(𝐺 2 ) < 25 𝑛1 + 25 𝑛2 = 25 𝑛, a contradiction. Hence, 𝐺 2 is a 25 -minimal graph. Using the same argument to show that 𝐺 1 ∉ Bdom , one can show that 𝐺 2 ∉ Bdom , and so 𝐺 2 ∈ Fdom ∪ Fsmall and 𝛾(𝐺 2 ) = 25 𝑛2 . The set 𝐷 1 can be extended to a dominating set of 𝐺 by adding a 𝛾-set of 𝐺 2 − 𝑣 2 to it, and so 𝛾(𝐺) ≤ |𝐷 1 | + 𝛾(𝐺 2 − 𝑣 2 ). Suppose that 𝐺 2 ∈ Fsmall . If 𝐺 2 = 𝐹1 and 𝑣 2 is a vertex of degree 2 in 𝐺 2 , then the edge joining 𝑣 2 to a vertex of degree 3 in 𝐺 2 joins two vertices of L but is not a bridge of 𝐺, a contradiction. Hence, 𝐺 2 ≠ 𝐹1 or 𝐺 2 = 𝐹1 and 𝑣 2 is a vertex of degree 3 in 𝐺 ′ . By Lemma 6.8(c), we have 𝛾(𝐺 2 − 𝑣 2 ) = 𝛾(𝐺 2 ) − 1, implying that 𝛾(𝐺) ≤ |𝐷 1 | + 𝛾(𝐺 2 ) − 1 = 25 𝑛1 + 25 𝑛2 − 1 < 25 𝑛, a contradiction. Hence, 𝐺 2 ∈ Fdom . Using similar arguments, one can show that 𝐺 1 ∈ Fdom . If 𝐺 1 = 𝐶5 and 𝐺 2 = 𝐶5 , then 𝐺 = 𝐷 (5, 5, 1) ∈ Fdom , a contradiction. Hence, renaming 𝐺 1 and 𝐺 2 if necessary, we may assume that 𝐺 2 ≠ 𝐶5 , and so 𝐺 2 has at least two units. Suppose that 𝑣 2 is not a link vertex in 𝐺 2 . We show that 𝛾(𝐺 2 − 𝑣 2 ) ≤ 𝛾(𝐺 2 ) − 1. Let 𝑈 be the unit in 𝐺 2 that contains 𝑣 2 . If 𝑈 is a key unit, let 𝑣 2′ be the link vertex in 𝑈, while if 𝑈 is a cycle unit, let 𝑣 2′ be a link vertex in 𝑈 that is adjacent to at least one other link vertex in 𝐺 2 (that belongs to a unit different from 𝑈). Since 𝐺 2 has at least two units, the existence of such a vertex 𝑣 2′ is guaranteed by the construction of 𝐺 2 ∈ Fdom . We note that (in both cases) the graph obtained from 𝑈 by deleting 𝑣 2 and 𝑣 2′ has a dominating vertex. Let 𝐷 2 consist of such a dominating vertex in 𝑈 − {𝑣 2 , 𝑣 2′ }, together with a 𝛾-set in 𝐺 2 − 𝑈 that contains all of its link vertices. The resulting set 𝐷 2 is a dominating set of 𝐺 2 that contains two vertices from every unit of 𝐺 2 different from 𝑈, and exactly one vertex from the unit 𝑈. Hence, 𝛾(𝐺 2 − 𝑣 2 ) ≤ |𝐷 2 | = 𝛾(𝐺 2 ) − 1 = 25 𝑛2 − 1. Since 𝑣 1 ∈ 𝐷 1 , vertex 𝑣 2 is dominated by 𝐷 1 , implying that 𝐷 1 ∪ 𝐷 2 is a dominating set of 𝐺. Hence, 𝛾(𝐺) ≤ |𝐷 1 | + |𝐷 2 | = 25 𝑛1 + 25 𝑛2 − 1 = 25 𝑛 − 1 < 25 𝑛, a contradiction. Hence, 𝑣 2 is a link vertex in 𝐺 2 .

Section 6.2. Bounds on the Domination Number

137

If 𝐺 1 = 𝐶5 , then 𝐺 ∈ Fdom , a contradiction. Hence, 𝐺 1 has at least two units. Interchanging the roles of 𝐺 1 and 𝐺 2 , the vertex 𝑣 1 is a link vertex in 𝐺 1 . Thus, 𝐺 ∈ Fdom , a contradiction. Recall that 𝐺 is a 25 -minimal graph of minimum order 𝑛 such that 𝐺 ∉ {𝐵1 , 𝐵2 , 𝐵4 } and 𝐺 ∉ Fdom ∪ Fsmall . By our earlier observations, 𝑛 ≥ 8 and |L| ≥ 3. Claim 6.16.2 The graph 𝐺 does not contain a path on five vertices with the two ends of the path not adjacent, and with one or both ends of the path of degree at least 3 and with the internal vertices all of degree 2 in 𝐺. Proof Suppose, to the contrary, that 𝑃 : 𝑣 𝑣 1 𝑣 2 𝑣 3 𝑣 4 is a path in 𝐺, where deg𝐺 (𝑣) ≥ 3, deg𝐺 (𝑣 4 ) ≥ 2 and deg𝐺 (𝑣 𝑖 ) = 2 for all 𝑖 ∈ [3], and where 𝑣 is not adjacent to 𝑣 4 . Let 𝐺 𝑃 be the graph of order 𝑛′ = 𝑛 − 3 ≥ 5 obtained from 𝐺 by deleting the set of vertices {𝑣 1 , 𝑣 2 , 𝑣 3 }. If deg𝐺 (𝑣 4 ) = 2 or if deg𝐺 (𝑣 4 ) ≥ 3 and 𝐺 is disconnected, then let 𝐺 ′ be obtained from 𝐺 𝑃 by adding the edge 𝑣𝑣 4 , while if deg𝐺 (𝑣 4 ) ≥ 3 and 𝐺 is connected, then let 𝐺 ′ = 𝐺 𝑃 . By construction, the graph 𝐺 ′ is edge-minimal with respect to the two conditions: (i) 𝐺 ′ is connected and (ii) 𝛿(𝐺 ′ ) ≥ 2. By Lemma 6.12, 𝛾(𝐺) ≤ 𝛾(𝐺 ′ ) + 1. If 𝛾(𝐺 ′ ) ≤ 25 𝑛′ , then 𝛾(𝐺) ≤ 25 (𝑛 − 3) + 1 < 25 𝑛, a contradiction. Hence, 𝛾(𝐺 ′ ) > 25 𝑛′ , implying that 𝐺 ′ ∈ {𝐵2 , 𝐵4 }, noting that 𝑛′ ≥ 5. If 𝐺 ′ = 𝐵4 , then |L| = 2, a contradiction. If 𝐺 ′ = 𝐵2 , then 𝐺 ′ = 𝐺 𝑃 and |L| = 3. In this case, rebuilding the graph 𝐺 back from the graph 𝐺 ′ by adding back the vertices 𝑣 1 , 𝑣 2 , 𝑣 3 and the edges of the path 𝑃, we produce a graph that is not a 25 -minimal graph, a contradiction. By Claim 6.16.1, the set L is an independent set. In particular, the large vertices at the ends of the linkage are not adjacent. Hence, as a consequence of Claim 6.16.2, we have the following structure of handles and linkages. Claim 6.16.3 The following hold in the graph 𝐺: (a) If 𝐺 contains a 𝑘-handle, then 𝑘 ∈ {3, 4, 5}. (b) If 𝐺 contains a 𝑘-linkage, then 𝑘 ∈ {1, 2}. We show next that 𝐺 contains no 4-handle or 5-handle. Claim 6.16.4 The graph 𝐺 does not contain a 4-handle. Proof Suppose, to the contrary, that 𝐺 contains a 4-handle 𝐻, given by 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 1 with deg𝐺 (𝑣 4 ) ≥ 3 and deg𝐺 (𝑣 𝑖 ) = 2 for 𝑖 ∈ [3]. Suppose that deg𝐺 (𝑣 4 ) ≥ 4. Let 𝐺 ′ be the graph of order 𝑛′ = 𝑛 − 3 ≥ 5 obtained from 𝐺 by deleting the vertices 𝑣 1 , 𝑣 2 , and 𝑣 3 . Every dominating set of 𝐺 ′ can be extended to a dominating set of 𝐺 by adding the vertex 𝑣 2 to it, and so 𝛾(𝐺) ≤ 𝛾(𝐺 ′ ) + 1. The graph 𝐺 ′ is edge-minimal with respect to the two conditions: (i) 𝐺 ′ is connected and (ii) 𝛿(𝐺 ′ ) ≥ 2. If 𝛾(𝐺 ′ ) ≤ 25 𝑛′ = 25 (𝑛 − 3), then 𝛾(𝐺) ≤ 25 (𝑛 − 3) + 1 < 25 𝑛, a contradiction. Hence, 𝛾(𝐺 ′ ) > 25 𝑛′ , implying that 𝐺 ′ ∈ {𝐵1 , 𝐵2 , 𝐵4 }. Since |L| ≥ 3, this in turn implies that 𝐺 = 𝐹3 ∈ Fsmall , a contradiction. Hence, deg𝐺 (𝑣 4 ) = 3. Let 𝑣 5 be the neighbor of 𝑣 4 that does not belong to the 4-handle 𝐻. Let 𝑁5 be the set of neighbors of 𝑣 5 different from 𝑣 4 , and so 𝑁5 = N𝐺 (𝑣 5 ) \ {𝑣 4 }.

138

Chapter 6. Upper Bounds in Terms of Minimum Degree

Suppose that deg𝐺 (𝑣 5 ) ≥ 3, and so |𝑁5 | ≥ 2. Let 𝐺 ′ be the graph obtained from 𝐺 − {𝑣 1 , 𝑣 2 , . . . , 𝑣 5 } by adding edges, if necessary, between vertices in 𝑁5 so that the resulting graph is edge-minimal with respect to the two conditions: (i) 𝐺 ′ is connected and (ii) 𝛿(𝐺 ′ ) ≥ 2. Let 𝐺 ′ have order 𝑛′ , and so 𝑛′ = 𝑛 − 5 ≥ 3. Every 𝛾-set of 𝐺 ′ can be extended to a dominating set of 𝐺 by adding the vertices 𝑣 2 and 𝑣 5 to it, and so 𝛾(𝐺) ≤ 𝛾(𝐺 ′ ) + 2. If 𝛾(𝐺 ′ ) < 52 𝑛′ , then 𝛾(𝐺) < 25 (𝑛 − 5) + 2 = 25 𝑛, a contradiction. Hence, 𝛾(𝐺 ′ ) ≥ 25 𝑛′ . By construction, the graph 𝐺 ′ is a 25 -minimal graph. Since 𝐺 ′ is not a counterexample, 𝐺 ′ ∈ {𝐵1 , 𝐵2 , 𝐵4 } ⊂ Bdom or 𝐺 ′ ∈ Fdom ∪ Fsmall . Let 𝑣 be an arbitrary neighbor of 𝑣 5 in 𝐺 ′ , and so 𝑣 ∈ 𝑁5 . Suppose that 𝐺 ′ ∈ {𝐵1 , 𝐵2 , 𝐵4 }. The set {𝑣 2 , 𝑣 5 } can be extended to a dominating set of 𝐺 by adding a 𝛾-set of 𝐺 ′ − 𝑣 to
it. Thus, by Proposition 6.7(c), we have 𝛾(𝐺) ≤ 2 + 𝛾(𝐺 ′ − 𝑣) = 2 + 𝛾(𝐺 ′ ) − 1 = 1 + 13 (𝑛′ + 2) = 1 + 13 (𝑛 − 3) = 13 𝑛 < 25 𝑛, a contradiction. Suppose that 𝐺 ′ ∈ Fsmall . If 𝐺 ′ = 𝐹1 and 𝑣 ′ is a vertex of degree 2 in 𝐺 ′ , then the graph 𝐺 is not a 25 -minimal graph, a contradiction. If 𝐺 ′ ≠ 𝐹1 or if 𝐺 ′ = 𝐹1 and 𝑣 ′ is a not a vertex of degree 2 in 𝐺 ′ , then by Lemma 6.8(c),
we have 𝛾(𝐺 ′ − 𝑣) = 𝛾(𝐺 ′ ) − 1, implying that 𝛾(𝐺) ≤ 2 + 𝛾(𝐺 ′ − 𝑣) = 2 + 𝛾(𝐺 ′ ) − 1 = 1 + 25 𝑛′ = 1 + 25 (𝑛 − 5) < 25 𝑛, a contradiction. Suppose that 𝐺 ′ ∈ Fdom . If it is possible to construct 𝐺 ′ in such a way that all the vertices of 𝑁5 are link vertices of 𝐺 ′ , then 𝐺 ∈ Fdom (with the vertex 𝑣 5 a link vertex in 𝐺), a contradiction. Hence, there is no construction of 𝐺 ′ where all the neighbors of 𝑣 5 in 𝐺 ′ are link vertices of 𝐺 ′ . Thus, we can choose the vertex 𝑣 so that 𝑣 is not a link vertex of 𝐺 ′ . If 𝐺 ′ = 𝐶5 , then |L| = 2, a contradiction. Hence, 𝐺 ′ contains at least two units. As shown in the proof of Claim 6.16.1, 𝛾(𝐺 ′ − 𝑣) ≤ 𝛾(𝐺 ′ ) − 1. The set {𝑣 2 , 𝑣 5 } can be extended to a dominating set of 𝐺 by adding a 𝛾-set of 𝐺 ′ − 𝑣 to it. Thus, by Proposition 6.7(c), we have
𝛾(𝐺) ≤ 2 + 𝛾(𝐺 ′ − 𝑣) = 2 + 𝛾(𝐺 ′ ) − 1 = 1 + 25 𝑛′ = 1 + 25 (𝑛 − 5) < 25 𝑛, a contradiction. Hence, deg𝐺 (𝑣 5 ) = 2, and so |𝑁5 | = 1. Let 𝑁5 = {𝑣 6 }. If deg𝐺 (𝑣 6 ) ≥ 3, then letting 𝐺 ′ = 𝐺 − {𝑣 1 , 𝑣 2 , . . . , 𝑣 5 }, we reach a contradiction, as above. Hence, deg𝐺 (𝑣 6 ) = 2. Let 𝑣 7 be the neighbor of 𝑣 6 different from 𝑣 5 . By Claim 6.16.3, the graph 𝐺 contains no 𝑘-linkage for 𝑘 ≥ 3, implying that deg𝐺 (𝑣 7 ) ≥ 3. We now consider the graph 𝐺 ′ = 𝐺 − {𝑣 1 , 𝑣 2 , . . . , 𝑣 6 } of order 𝑛′ = 𝑛 − 6. Every 𝛾-set of 𝐺 ′ can be extended to dominating set of 𝐺 by adding the set {𝑣 2 , 𝑣 5 } to it, and so 𝛾(𝐺) ≤ 2 + 𝛾(𝐺 ′ ). If 𝛾(𝐺 ′ ) ≤ 25 𝑛′ , then 𝛾(𝐺) ≤ 2 + 25 𝑛′ = 2 + 25 (𝑛 − 6) < 25 𝑛, a contradiction. Hence, 𝛾(𝐺 ′ ) > 25 𝑛′ , implying that 𝐺 ′ ∈ {𝐵1 , 𝐵2 , 𝐵4 }. Since |L| ≥ 3, this in turn implies that 𝐺 ′ = 𝐵2 . Thus, 𝑛 = 13 and 𝛾(𝐺) = 5 < 25 𝑛, a contradiction. Claim 6.16.5 The graph 𝐺 does not contain a 5-handle. Proof Suppose, to the contrary, that 𝐺 contains a 5-handle 𝐻, given by 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑣 1 with 𝑣 5 as its (unique) vertex of degree at least 3. Thus, each vertex 𝑣 𝑖 has degree 2 in 𝐺 for 𝑖 ∈ [4]. By supposition, deg𝐺 (𝑣 5 ) ≥ 3. Let 𝑁5 be the set of neighbors of 𝑣 5 that do not belong to the 5-handle 𝐻. If deg𝐺 (𝑣 5 ) ≥ 4, then |𝑁5 | ≥ 2 and proceeding as in the proof of Claim 6.16.4, we obtain a contradiction. Hence, deg𝐺 (𝑣 5 ) = 3, and so |𝑁5 | = 1. Let 𝑁5 = {𝑣 6 }. Proceeding once again exactly as in the proof of Claim 6.16.4 we obtain a contradiction.

Section 6.2. Bounds on the Domination Number

139

By Claims 6.16.3, 6.16.4, and 6.16.5, we have the following structural result in 𝐺. Claim 6.16.6 Every handle in 𝐺, if any, is a 3-handle, and every linkage in 𝐺 is a 1-linkage or a 2-linkage. Let |L| = ℓ and |S| = 𝑠, and so 𝑛 = ℓ+𝑠. By Claim 6.16.6, the set L is a dominating set of 𝐺. Thus, since 𝐺 is a 52 -minimal graph, ℓ = |L| ≥ 𝛾(𝐺) ≥ 25 𝑛 = 25 (ℓ + 𝑠), and so 3ℓ ≥ 2𝑠. However, counting edges between L and S, we have 3ℓ ≤ 2𝑠, noting that each vertex in L is adjacent to at least three vertices in S and each vertex in S is adjacent to at most two neighbors in L. Consequently, 3ℓ = 2𝑠. Thus, ℓ = 23 𝑠 = 23 (𝑛 − ℓ), or equivalently, ℓ = 25 𝑛. Further, each vertex in L has degree exactly 3 and the set S is an independent set, and so every edge is incident with exactly one vertex of degree 2. This in turn implies that 𝐺 has no 3-handles and that every linkage in 𝐺 is a 1-linkage. Thus, 𝐺 is a bipartite graph with partite sets L and S, where each vertex in L has degree 3 and each vertex in S has degree 2. Since 𝑛 ≥ 8, we note that 𝐺 ≠ 𝐾2,3 = 𝐹1 and |L| ≥ 3. Hence, there exist two
vertices 𝑢 and 𝑣 in L with exactly one common neighbor 𝑤. The set L \ {𝑢, 𝑣} ∪ {𝑤} is a dominating set of 𝐺 of cardinality |L| − 1 = 25 𝑛 − 1 < 25 𝑛, a contradiction. This completes the proof of Theorem 6.16. As an immediate consequence of Theorem 6.16, we have the following result. Corollary 6.17 ([586]) If 𝐺 is a 25 -minimal graph of order 𝑛 > 10, then 𝐺 ∈ Fdom . We are now in a position to state the McCuaig-Shepherd result. Theorem 6.18 ([586]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then 𝛾(𝐺) ≤ 25 𝑛, unless 𝐺 is one of the seven exceptional graphs in the family Bdom . Proof Let 𝐺 be a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2. Let 𝐺 ′ be obtained from 𝐺 by deleting edges, if necessary, so that the resulting graph is edge minimal with respect to the two conditions: (i) 𝐺 ′ is connected and (ii) 𝛿(𝐺 ′ ) ≥ 2. Since removing edges from a graph cannot decrease its domination number, 𝛾(𝐺) ≤ 𝛾(𝐺 ′ ). If 𝐺 ′ is not a 25 -minimal graph, then 𝛾(𝐺) ≤ 𝛾(𝐺 ′ ) < 25 𝑛. If 𝐺 ′ is a 25 -minimal graph, then by Theorem 6.16, 𝐺 ′ ∈ {𝐵1 , 𝐵2 , 𝐵4 } or 𝐺 ′ ∈ Fdom ∪ Fsmall . If 𝐺 ′ ∈ Fdom ∪ Fsmall , then 𝑛 ≥ 5 and 𝛾(𝐺 ′ ) = 25 𝑛, implying that 𝛾(𝐺) ≤ 25 𝑛. If 𝐺 ′ = 𝐵1 , then 𝑛 = 4 and either 𝛾(𝐺) = 1 < 25 𝑛 or 𝐺 = 𝐺 ′ and 𝛾(𝐺) = 2 = 15 (2𝑛 + 2). If 𝐺 ′ ∈ {𝐵2 , 𝐵4 }, then 𝑛 = 7 and 𝛾(𝐺 ′ ) = 3 = 15 (2𝑛 + 1). Further, in this case either 𝛾(𝐺) ≤ 2 < 25 𝑛 or 𝐺 ∈ Bdom \ {𝐵1 }, in which case 𝛾(𝐺) = 15 (2𝑛 + 1). Thus, 𝛾(𝐺) ≤ 25 𝑛, unless 𝐺 ∈ Bdom . Let Gdom be the family of all graphs that can be obtained from a graph in the family Fdom by adding edges, including the possibility of none, joining link vertices. For example, for the graph 𝐹 ∈ Fdom shown in Figure 6.5(a) with the link vertices of 𝐹 given by the highlighted vertices, a graph 𝐺 ∈ Gdom obtained from 𝐹 is shown in Figure 6.5(b). We note that Fdom ⊂ Gdom . Theorem 6.19 ([79]) If 𝐺 is a connected graph of order 𝑛 > 10 with 𝛿(𝐺) ≥ 2, then 𝛾(𝐺) ≤ 25 𝑛, with equality if and only if 𝐺 ∈ Gdom .

Chapter 6. Upper Bounds in Terms of Minimum Degree

140

(a) 𝐹 ∈ Fdom

(b) 𝐺 ∈ Gdom

Figure 6.5 A graph 𝐺 ∈ Gdom obtained from a graph 𝐹 ∈ Fdom Proof Let 𝐺 ∈ Gdom have order 𝑛 > 10. Every dominating set of 𝐺 contains at least two vertices from each unit of 𝐺. However, there exists a dominating set of 𝐺 that contains exactly two vertices from each unit of 𝐺. Thus, 𝛾(𝐺) = 25 𝑛. Conversely, suppose that 𝐺 is a connected graph of order 𝑛 > 10 with 𝛿(𝐺) ≥ 2 satisfying 𝛾(𝐺) = 25 𝑛. Let 𝐹 be obtained from 𝐺 by deleting edges, if necessary, so that the resulting graph is edge minimal with respect to the two conditions: (i) 𝐹 is connected and (ii) 𝛿(𝐹) ≥ 2. If 𝐹 is not a 25 -minimal graph, then 𝛾(𝐺) ≤ 𝛾(𝐹) < 25 𝑛, a contradiction. Hence, 𝐹 is a 25 -minimal graph of order 𝑛 > 10. Thus, by Corollary 6.17, 𝐹 ∈ Fdom and 𝛾(𝐹) = 25 𝑛. If there is an edge 𝑒 ∈ 𝐸 (𝐺) \ 𝐸 (𝐹) such that there is no construction of 𝐹 in which both ends of 𝑒 are link vertices, then analogous methods used in the proof of Claim 6.16.1 show that 𝛾(𝐺) < 𝛾(𝐹) = 25 𝑛, a contradiction. Hence, 𝐺 can be obtained from the graph 𝐹 ∈ Fdom by adding edges, including the possibility of none, joining link vertices of 𝐹, implying that 𝐺 ∈ Gdom . As remarked in [477], there are infinitely many 2-connected graphs that achieve equality in the bound of Theorem 6.18. One such family can be constructed as follows: Let 𝑘 ≥ 2 be an integer and let F2conn be the family of all graphs that can be obtained from a 2-connected graph 𝐹 of order 2𝑘 that contains a perfect matching 𝑀 as follows. Replace each edge 𝑢𝑣 ∈ 𝑀 by a 5-cycle containing 𝑢 and 𝑣 as nonadjacent vertices on the cycle. Let 𝐺 denote the resulting graph of order 𝑛 = 5𝑘. Then, 𝛾(𝐺) = 2𝑘 = 25 𝑛. A graph in the family F2conn with 𝑘 = 4 that is obtained from an 8-cycle 𝐹 is shown in Figure 6.6.

𝑢

𝑣

Figure 6.6 A graph in the family F2conn

6.2.3

Minimum Degree Three

In 1996 Reed [655] proved that the 25 -bound in Theorem 6.18 can be improved to a 3 8 -bound if we restrict the minimum degree to be at least 3. In this section, we present

Section 6.2. Bounds on the Domination Number

141

Reed’s proof of this classical result. For this purpose, recall that a vertex-disjoint path cover or just path cover, abbreviated vdp-cover, of a graph 𝐺 is set of vertex-disjoint paths 𝑄 1 , 𝑄 2 , . . . , 𝑄 𝑘 (not necessarily induced) that cover the vertices of 𝐺, that is, 𝑉=

𝑘 Ø

𝑉 (𝑄 𝑖 ).

𝑖=1

Let |𝑃| denote the number of vertices in a path 𝑃. A path 𝑃 is called a 0-, 1- or 2-path if |𝑃| is congruent to 0, 1, or 2 modulo 3, respectively. Thus, if 𝑃 is an 𝑖-path where 𝑖 ∈ [2] 0 , then |𝑃| ≡ 𝑖 (mod 3). If 𝑃 is a (𝑢, 𝑣)-path, then we call 𝑢 and 𝑣 the ends of the path 𝑃. Every vertex of a path different from its ends we call an internal vertex of the path. We now present the main ideas of the proof of Reed’s result. However, we omit some of the more technical details of the proof. Theorem 6.20 ([655]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾(𝐺) ≤ 83 𝑛. Proof Sketch Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺) ≥ 3. Let 𝑆 be a vdp-cover of 𝐺 and let 𝑆𝑖 be the set of 𝑖-paths in 𝑆 for 𝑖 ∈ [2] 0 . If 𝑣 is an internal vertex on a path 𝑃 and 𝑃 − 𝑣 consists of an 𝑖-path and a 𝑗-path, then we call 𝑣 an (𝑖, 𝑗)-vertex of 𝑃. An end 𝑣 of a path 𝑃 ∈ 𝑆 is an out-endvertex of the path 𝑃 if 𝑣 has a neighbor which is not on 𝑃. If 𝑃 ∈ 𝑆2 , and so 𝑃 is a 2-path, then an end of 𝑃 is a (2, 2)-endvertex of 𝑃 if it is not an out-endvertex and is adjacent to a (2, 2)-vertex of 𝑃. We define an optimal vdp-cover of 𝐺 to be a vdp-cover 𝑆 for which the following hold: (a) |𝑆| is minimized. (b) Subject to (a), 2|𝑆1 | + |𝑆2 | is minimized. (c) Subject to (b), |𝑆2 | is minimized. Í (d) Subject to (c), Í𝑃 ∈𝑆0 |𝑃| is minimized. (e) Subject to (d), 𝑃 ∈𝑆1 |𝑃| is minimized. (f) Subject to (e), the total number of out-endvertices is maximized. We proceed further with the following properties of an optimal vdp-cover 𝑆. Claim 6.20.1 If 𝑆 is an optimal vdp-cover of 𝐺 and 𝑥 is an out-endvertex of a path 𝑃 ∈ 𝑆1 ∪ 𝑆2 that is adjacent to a vertex 𝑦 in some path 𝑄 ∈ 𝑆 distinct from 𝑃, then the following hold: (a) 𝑄 is not a 1-path, that is, 𝑄 ∈ 𝑆0 ∪ 𝑆2 . (b) If 𝑄 ∈ 𝑆0 , then 𝑦 is a (1, 1)-vertex of 𝑄. (c) If 𝑄 ∈ 𝑆2 , then 𝑦 is a (2, 2)-vertex of 𝑄. Proof Sketch Let 𝑄 = 𝑄 𝑖 𝑦 𝑄 𝑗 , where 𝑄 𝑖 is an 𝑖-path and 𝑄 𝑗 is a 𝑗-path, and so 𝑦 is an (𝑖, 𝑗)-vertex of the path 𝑄. Interchanging 𝑄 𝑖 and 𝑄 𝑗 if necessary, we may assume 𝑖 ≤ 𝑗. By Condition (a) of the optimal vdp-cover 𝑆, vertex 𝑦 is not an end of 𝑄, and so 𝑦 is an internal vertex of 𝑄. Let 𝑆 ′ and 𝑆 ′′ be the vdp-covers obtained from 𝑆 as follows:
𝑆 ′ = 𝑆 \ {𝑃, 𝑄} ∪ {𝑄 𝑖 , 𝑃 𝑦 𝑄 𝑗 },
𝑆 ′′ = 𝑆 \ {𝑃, 𝑄} ∪ {𝑃 𝑦 𝑄 𝑖 , 𝑄 𝑗 }.

142

Chapter 6. Upper Bounds in Terms of Minimum Degree

(a) Suppose, to the contrary, that 𝑄 ∈ 𝑆1 . Then, 𝑦 is either a (0, 0)-vertex or a (1, 2)-vertex of 𝑄. If 𝑃 ∈ 𝑆1 and 𝑦 is a (0, 0)-vertex, then 𝑆 ′ replaces two 1-paths with a 0-path and a 2-path, while if 𝑃 ∈ 𝑆1 and 𝑦 is a (1, 2)-vertex, then 𝑆 ′′ replaces two 1-paths with a 0-path and a 2-path. In both cases, we contradict the choice of 𝑆 as an optimal vdp-cover noting that we violate Condition (b). If 𝑃 ∈ 𝑆2 and 𝑦 is a (0, 0)-vertex, then 𝑆 ′ replaces a 1-path and a 2-path with two 0-paths, once again violating Condition (b) of the optimal vdp-cover 𝑆. If 𝑃 ∈ 𝑆2 and 𝑦 is a (1, 2)-vertex, then 𝑆 ′ has lengthened a 2-path and shortened a 1-path, violating Condition (e) of the optimal vdp-cover 𝑆. Both cases therefore contradict the choice of 𝑆 as an optimal vdp-cover. This proves part (a). (b) Suppose that 𝑄 ∈ 𝑆0 . Then, 𝑦 is either a (0, 2)-vertex or a (1, 1)-vertex of 𝑄. Suppose, to the contrary, that 𝑦 is a (0, 2)-vertex of 𝑄. If 𝑃 ∈ 𝑆1 , then 𝑆 ′ has lengthened a 1-path and shortened a 0-path, while if 𝑃 ∈ 𝑆2 , then 𝑆 ′ has lengthened a 2-path and shortened a 0-path. In both cases, we violate Condition (d) of the optimal vdp-cover 𝑆. Hence, 𝑦 is a (1, 1)-vertex of 𝑄. This proves part (b). (c) Suppose that 𝑄 ∈ 𝑆2 . Then, 𝑦 is either a (0, 1)-vertex or a (2, 2)-vertex of 𝑄. Suppose, to the contrary, that 𝑦 is a (0, 1)-vertex of 𝑄. If 𝑃 ∈ 𝑆1 , then 𝑆 ′ replaces a 1-path and a 2-path with two 0-paths, violating Condition (b) of the optimal vdp-cover 𝑆. If 𝑃 ∈ 𝑆2 , then 𝑆 ′ replaces two 2-paths with a 0-path and a 1-path, violating Condition (c) of the optimal vdp-cover 𝑆. Hence, 𝑦 is a (2, 2)-vertex of 𝑄. This proves part (c) and completes the proof of Claim 6.20.1. For each 1-path 𝑃 ∈ 𝑆 that has an out-endvertex 𝑥, we choose a vertex 𝑦 not in 𝑃 that is adjacent to 𝑥. For each 2-path 𝑃 ∈ 𝑆 that has two out-endvertices, we choose for each out-endvertex 𝑥 of 𝑃 a vertex 𝑦 not in 𝑃 that is adjacent to 𝑥. For each 2-path 𝑃 ∈ 𝑆 on five vertices that induces the graph 𝐹 shown in Figure 6.7 and which has precisely one out-endvertex, namely the vertex 𝑥 of degree 2 in 𝐹, we choose a vertex 𝑦 not in 𝑃 that is adjacent to the out-endvertex 𝑥 of 𝑃. In all the above cases, we designate the vertex 𝑦 as an acceptor for 𝑥, and we call 𝑦 an acceptor for the path 𝑃 associated with the vertex 𝑥.

𝑥 Figure 6.7 The graph 𝐹 We call a path in the optimal vdp-cover 𝑆 of the graph 𝐺 an accepting path if at least one of its vertices was designated as an acceptor for some path in 𝑆. Let 𝐴 ⊆ 𝑆 be the set of accepting 2-paths in 𝑆. For each out-endvertex 𝑥 of a path 𝑃 in 𝐴 for which we have not yet chosen an acceptor (this occurs because the path has only one out-endvertex), we choose a vertex adjacent to this out-endvertex in 𝐺 − 𝑉 (𝑃) and designate this vertex to be an acceptor for 𝑥. If this new acceptor is on a previously non-accepting 2-path 𝑃′ , then we add 𝑃′ to 𝐴. Continue this process until there is an acceptor for every out-endvertex of the paths in 𝐴. Furthermore, for

Section 6.2. Bounds on the Domination Number

143

every (2, 2)-endvertex 𝑥 of a path 𝑃 ∈ 𝐴, we choose a (2, 2)-vertex 𝑦 of 𝑃 which is adjacent to 𝑥 and designate it as an in-acceptor for 𝑥. By Claim 6.20.1, every acceptor vertex and every in-acceptor vertex that belongs to an accepting 2-path is a (2, 2)-vertex. Each accepting 2-path 𝑃 ∈ 𝑆 can therefore be written in the form 𝑄 1 𝑄 2 𝑄 3 , where 𝑄 1 and 𝑄 3 are both 1-paths containing no acceptors or in-acceptors, and are maximal with respect to this property. By Claim 6.20.1 and our earlier observations, the second and penultimate (that is second to last) vertices of 𝑄 2 are acceptors or in-acceptors. The paths 𝑄 1 and 𝑄 3 are called tips of 𝑃, while 𝑄 2 is called the central path of 𝑃. We next construct a dominating set 𝐷 of 𝐺 as follows. Initially, we let 𝐷 = ∅. We now build our dominating set 𝐷 in the following fashion: Step 1. Let 𝐷 0 be the set of all (1, 1)-vertices of 0-paths, and add all vertices in 𝐷 0 to the set 𝐷. Step 2. Let 𝐷 2 be the set of all (2, 2)-vertices of accepting 2-path 𝑃 that belong to the central path of 𝑃, and add all vertices in 𝐷 2 to the set 𝐷. To illustrate Steps 1 and 2, the (2, 2)-vertices of a 2-path 𝑃 and the three (1, 1)vertices of a 0-path 𝑃 are illustrated in Figure 6.8(a) and (b), respectively, by the highlighted vertices. By Claim 6.20.1, if 𝑥 is an out-endvertex of a path 𝑃 ∈ 𝑆1 ∪ 𝑆2 , then the vertex 𝑥 is dominated by the set 𝐷 0 ∪ 𝐷 2 . Thus, the set 𝐷 0 ∪ 𝐷 2 dominates all the out-endvertices of paths in 𝑆1 ∪ 𝑆2 . (2, 2)

(2, 2) (a) A 2-path

(1, 1)

(1, 1)

(1, 1)

(b) A 0-path

Figure 6.8 A 2-path and a 0-path   Step 3. For each 1-path with at least one out-endvertex, choose | 𝑃3 | vertices of 𝑃 which dominate all the vertices of 𝑃, except possibly for the out-endvertex 𝑥 of 𝑃 that is adjacent to the acceptor of 𝑃. As observed earlier, vertex 𝑥 is dominated   by the set 𝐷 0 ∪ 𝐷 2 . We now add these | 𝑃3 | vertices of 𝑃 to the set 𝐷. Step 4. For each non-accepting 2-path 𝑃 with both ends of 𝑃 either an outendvertex or a (2,2)-endvertex, we add all (2, 2)-vertices of 𝑃 to the set 𝐷. We  note that there are | 𝑃3 | such (2, 2)-vertices and these vertices dominate all vertices of 𝑃, except possibly for the ends of 𝑃 which are dominated by their acceptors or in-acceptors in the set 𝐷 0 ∪ 𝐷 2 . Step 5. For each 2-path 𝑃 on five vertices whose vertices induce the graph 𝐹 in Figure 6.6 and that has precisely one out-endvertex, namely the vertex 𝑥 of degree 2 in 𝐹, we add to the set 𝐷 a vertex in 𝐹 not adjacent to the out-endvertex 𝑥 of 𝐹. We note that such an added vertex dominates all vertices of 𝐹, except for the vertex 𝑥

144

Chapter 6. Upper Bounds in Terms of Minimum Degree

which is dominated by its acceptor that belongs to the set 𝐷 0 ∪ 𝐷 2 . Suppose that the 2-path 𝐹 is an accepting 2-path with acceptor 𝑦 for some path 𝑃′ associated with an out-endvertex 𝑥 ′ . In this case, 𝑦 is a (2, 2)-vertex of 𝐹 and so, by the structure of 𝐹, we could replace the two paths 𝑃′ and 𝐹 with one new path (that contains the edge 𝑥 ′ 𝑦, and has the vertex 𝑥 as an out-endvertex). This violates Condition (a) of the optimal vdp-cover 𝑆. Hence, the 2-path 𝐹 is a non-accepting 2-path. Step 6. For each 1-path 𝑃 with no out-endvertex, choose a 𝛾-set of 𝐺 [𝑉 (𝑃)] and add the vertices of this set to 𝐷. Let 𝐹 be the graph in Figure 6.7. For each non-accepting 2-path 𝑃 with at most one out-endvertex, which either does not induce the graph 𝐹 or induces the graph 𝐹 but does not have the vertex of degree 2 in 𝐹 as its out-endvertex, choose a 𝛾-set of 𝑃 and add the vertices of this set to 𝐷. In the   above cases, we note that the 𝛾-set of 𝐺 [𝑉 (𝑃)] has at most | 𝑃3 | vertices. Step 7. For each tip 𝑃1 of an accepting 2-path 𝑃, if the common end 𝑥 of 𝑃1 and 𝑃 is an out-endvertex or a (2, 2)-endvertex, then add to the  set 𝐷 all (2, 2)-vertices of 𝑃 that belong to the tip 𝑃1 . We note that there are | 𝑃3 | such (2, 2)-vertices and these vertices dominate all vertices of 𝑃1 , except possibly for 𝑥, which is dominated by the set 𝐷 0 ∪ 𝐷 2 . Step 8. For each tip 𝑃1 of an accepting 2-path 𝑃, if the common end 𝑥 of 𝑃1 and 𝑃 is neither an out-endvertex nor a (2, 2)-endvertex, then choose a 𝛾-set of 𝐺 [𝑉 (𝑃1 )] and add the vertices of this set to 𝐷. We note that the 𝛾-set of 𝐺 [𝑉 (𝑃)] has at most  |𝑃|  3 vertices. We show next that the set 𝐷 constructed above is a dominating set of 𝐺. Claim 6.20.2 The set 𝐷 constructed by Steps 1–8 is a dominating set of 𝐺. Proof Sketch As observed earlier, every acceptor belongs to the set 𝐷 0 ∪ 𝐷 2 , and therefore belongs to the set 𝐷. By construction, the set 𝐷 0 chosen in Step 1 dominates the vertices on all 0-paths. Hence, the vertices of the 0-paths are dominated by 𝐷. The vertices chosen in Step 3 ensure that all vertices of a 1-path 𝑃 with an out-endvertex are dominated by 𝐷 since all acceptors are in 𝐷, while the vertices chosen in Step 6 ensure that all vertices of a 1-path 𝑃 with no out-endvertex are dominated by 𝐷. Hence, the vertices of the 1-paths are dominated by 𝐷. The vertices chosen in Steps 4, 5, and 6 ensure that all vertices of non-accepting 2-paths 𝑃 are dominated by 𝐷, since all acceptors are in 𝐷. Hence, the vertices of all non-accepting 2-paths are dominated by 𝐷. The vertices in 𝐷 2 chosen in Step 2 dominate all vertices that belong to a central path of accepting 2-paths, while the vertices chosen in Steps 7 and 8 ensure that all vertices that belong to the tips of accepting 2-paths are dominated by 𝐷. Hence, the vertices of all accepting 2-paths are dominated by 𝐷. Thus, the vertices of the 2-paths are dominated by 𝐷. By Claim 6.20.2, the set 𝐷 is a dominating set of 𝐺. It remains to show that |𝐷| ≤ 38 𝑛. For this purpose, we define the following sets: • 𝑂 1 : the set of 1-paths 𝑃 which either have an out-endvertex or contain a dominating   set of cardinality | 𝑃3 | . • 𝑂 2 : the set of non-accepting 2-paths which either have two out-endvertices or a   dominating set of cardinality | 𝑃3 | and all 2-paths which induce the graph 𝐹 of

Section 6.2. Bounds on the Domination Number

145

Figure 6.7 and which have precisely one out-endvertex, namely the vertex 𝑥 of degree 2 in 𝐹. • 𝐼1 : the set of 1-paths not in 𝑂 1 . • 𝐼2 : the set of non-accepting 2-paths not in 𝑂 2 . • 𝐸: a tip 𝑇 of an accepting 2-path 𝑃 is in 𝐸 if and only if the corresponding end of 𝑃 is neither  an out-endvertex nor a (2, 2)-endvertex, and 𝑇 cannot be dominated by the | 𝑃3 | vertices of the central path of 𝑃. • 𝑊: the set of (2, 2)-endvertices of accepting 2-paths for which we have chosen an in-acceptor. Recall that 𝐴 ⊆ 𝑆 is the set of accepting 2-paths in 𝑆. The cardinality of the dominating set 𝐷 is given by ∑︁ |𝑃| − 1 ∑︁ |𝑃| − 2 ∑︁ |𝑃| + 2 |𝐷| = + + 3 3 3 𝑃∈ 𝐼1 𝑃∈𝑂1 𝑃∈𝑂2 ∑︁ |𝑃| + 1 ∑︁ |𝑃| ∑︁ |𝑃| − 2 + + + + |𝐸 |. 3 3 3 𝑃∈ 𝐼 𝑃∈𝑆 𝑃∈ 𝐴 2

0

Equivalently, |𝐷| =

𝑛 3

− 13 |𝑂 1 | − 23 |𝑂 2 | + 23 |𝐼1 | + 13 |𝐼2 | − 23 | 𝐴| + |𝐸 |.

(6.1)

Note that each accepting 2-path corresponds to an out-endvertex of some path in 𝑂 1 ∪ 𝑂 2 or to an out-endvertex of an accepting 2-path of 𝐴 which is not in 𝐸 ∪ 𝑊. Thus, | 𝐴| ≤ |𝑂 1 | + 2|𝑂 2 | + 2| 𝐴| − |𝐸 | − |𝑊 |, or equivalently, |𝐸 | ≤ |𝑂 1 | + 2|𝑂 2 | + | 𝐴| − |𝑊 |. (6.2) We note that |𝐸 | ≤ 2| 𝐴| − |𝑊 |. Hence, by Inequality (6.2), |𝐸 | ≤ 23 |𝑂 1 | + 43 |𝑂 2 | + 43 | 𝐴| − |𝑊 |, or equivalently, − 23 | 𝐴| ≤ 13 |𝑂 1 | + 23 |𝑂 2 | − 12 |𝐸 | − 12 |𝑊 |.

(6.3)

Substituting Inequality (6.3) into Equation (6.1), |𝐷 | ≤

𝑛 3

+ 23 |𝐼1 | + 13 |𝐼2 | + 12 |𝐸 | − 12 |𝑊 |.

To each element of 𝐸 there corresponds an accepting 2-path 𝑃𝑇 such that 𝑇 is the tip of 𝑃𝑇 . With this notation, we define 𝐸 ′ ⊆ 𝐸 as follows: 𝐸 ′ = {𝑇 ∈ 𝐸 : the end of 𝑃𝑇 not in 𝑇 is not an element of 𝑊 }. Clearly, |𝐸 ′ | ≥ |𝐸 | − |𝑊 |, and so |𝐷| ≤

𝑛 3

+ 23 |𝐼1 | + 13 |𝐼2 | + 12 |𝐸 ′ |.

(6.4)

We proceed further with the following two claims. We omit the proofs of these claims, which are technical in parts, and can also be checked using a computer proof.

Chapter 6. Upper Bounds in Terms of Minimum Degree

146

Claim 6.20.3 The following hold: (a) Each 1-path 𝑃 ∈ 𝐼1 satisfies |𝑃| ≥ 16. (b) Each 2-path 𝑃 ∈ 𝐼2 satisfies |𝑃| ≥ 8. Claim 6.20.4 If a path 𝑇 is a tip in 𝐸 ′ of an accepting 2-path 𝑃, and 𝑐 0 is the end of the central path in 𝑃 that is adjacent to an end of 𝑇, then the following hold: (a) If 𝐺 is a cubic graph, then either |𝑇 | ≥ 10 or 𝐺 is isomorphic to the graph 𝐻1 shown in Figure 6.9. (b) If 𝐺 is not a cubic graph, then |𝑇 | ≥ 7, with equality if and only if the subgraph of 𝐺 induced by 𝑉 (𝑇) ∪ {𝑐 0 } is isomorphic to the graph 𝐻1 , where in addition the vertex 𝑐 0 is the only vertex in 𝐻1 whose degree in 𝐺 is greater than 3. (c) If 𝐺 is not a cubic graph, then |𝑇 | ≥ 10, with the exception of at most two tips 𝑇 in 𝐸 ′ (that satisfy |𝑇 | = 7).

𝑐0

Figure 6.9 The graph 𝐻1 in the statement of Claim 6.20.4

It follows from Claim 6.20.3(a) and (b) that ∑︁ ∑︁ |𝑃| ≥ 16|𝐼1 | and |𝑃| ≥ 8|𝐼2 |. 𝑃∈ 𝐼1

(6.5)

𝑃∈𝐼2

We note that the central path of an accepting 2-path has at least three vertices. Assume first that 𝐺 is a cubic graph. If 𝐺 = 𝐻1 , then 𝛾(𝐺) = 3 = 38 𝑛, as desired. Hence, we may assume that 𝐺 ≠ 𝐻1 . Each element 𝑇 of 𝐸 ′ is the tip of some accepting 2-path 𝑃, and by Claim 6.20.4(a) satisfies |𝑇 | ≥ 10. If the other tip 𝑇 ′ of 𝑃 is also in 𝐸 ′ , then by Claim 6.20.4, we have |𝑃| ≥ 23. Otherwise, |𝑃| ≥ 14, or equivalently, |𝑃| − 1 ≥ 13. In addition, we know that there is some out-endvertex whose acceptor is on 𝑃. For an accepting 2-path 𝑃, let 𝑡 (𝑃) = (the number of tips of 𝑃 in 𝐸 ′ ) 𝜓1 (𝑃) = (the number of out-endvertices whose acceptor is on 𝑃) 𝜓2 (𝑃) = (the number of out-endvertices of 𝑃). Hence, 𝜓1 (𝑃) ≥ 1 and 𝑡 (𝑃) ∈ {0, 1, 2}. If 𝑡 (𝑃) = 0, then |𝑃| ≥ 5 and 𝜓2 (𝑃) ≤ 2. If 𝑡 (𝑃) = 1, then |𝑃| ≥ 14 and 𝜓2 (𝑃) ≤ 1. If 𝑡 (𝑃) = 2, then |𝑃| ≥ 23 and 𝜓2 (𝑃) = 0. Thus, for every accepting 2-path 𝑃, |𝑃| + 𝜓1 (𝑃) − 𝜓2 (𝑃) ≥ 12𝑡 (𝑃).

(6.6)

Section 6.2. Bounds on the Domination Number

147

Let 𝜑( 𝐴) = (the number of out-endvertices of paths in 𝐴) 𝜑(𝑂 1 ) = (the number of out-endvertices of paths in 𝑂 1 that have a designated acceptor) 𝜑(𝑂 2 ) = (the number of out-endvertices of paths in 𝑂 2 that have a designated acceptor) 𝜑(𝑆) = (the number of out-endvertices of paths in 𝑆 that have a designated acceptor). Then, ∑︁

∑︁

𝜓1 (𝑃) = 𝜑(𝑆),

𝑃∈ 𝐴

𝜓2 (𝑃) = 𝜑( 𝐴),

∑︁

and

𝑡 (𝑃) = |𝐸 ′ |.

𝑃∈ 𝐴

𝑃∈ 𝐴

Thus, by Inequality (6.6), summing over all paths 𝑃 ∈ 𝐴, we have   ∑︁ |𝑃| + 𝜑(𝑆) − 𝜑( 𝐴) ≥ 12|𝐸 ′ |.

(6.7)

𝑃∈ 𝐴

For each path 𝑃 ∈ 𝑆, let 𝜑(𝑃) = (the number of out-endvertices of 𝑃 that have a designated acceptor). For each path 𝑃 ∈ 𝑂 1 ∪ 𝑂 2 , |𝑃| ≥ 𝜑(𝑃). Thus, ∑︁ |𝑃| ≥ 𝜑(𝑂 1 ) + 𝜑(𝑂 2 ). (6.8) 𝑃∈𝑂1 ∪𝑂2

Since 𝜑( 𝐴) + 𝜑(𝑂 1 ) + 𝜑(𝑂 2 ) = 𝜑(𝑆), we have by Inequalities (6.7) and (6.8) that ∑︁ (6.9) |𝑃| ≥ 𝜑( 𝐴) + 𝜑(𝑂 1 ) + 𝜑(𝑂 2 ) − 𝜑(𝑆) + 12|𝐸 ′ | = 12|𝐸 ′ |. 𝑃∈ 𝐴∪𝑂1 ∪𝑂2

Since 𝑆 = 𝑂 1 ∪ 𝑂 2 ∪ 𝐼1 ∪ 𝐼2 ∪ 𝐴, we have by Inequalities (6.5) and (6.9) that ∑︁ 𝑛= |𝑃| ≥ 16|𝐼1 | + 8|𝐼2 | + 12|𝐸 ′ |, 𝑃 ∈𝑆

or equivalently, 2 3 |𝐼1 |

+ 13 |𝐼2 | + 12 |𝐸 ′ | ≤

𝑛 24 .

(6.10)

Substituting Inequality (6.10) into Inequality (6.4) yields 𝛾(𝐺) = |𝐷| ≤

𝑛 3

+ 23 |𝐼1 | + 13 |𝐼2 | + 12 |𝐸 ′ | ≤

𝑛 3

+

𝑛 24

= 38 𝑛.

This completes the proof of Theorem 6.20 in the case when 𝐺 is a cubic graph. If 𝐺 is not a cubic graph, then analogous arguments as above show, using Claim 6.20.4, that ∑︁ |𝑃| ≥ 12|𝐸 ′ | − 6 𝑃∈ 𝐴∪𝑂1 ∪𝑂2

148

Chapter 6. Upper Bounds in Terms of Minimum Degree

and 𝑛+6=

∑︁

|𝑃| ≥ 16|𝐼1 | + 8|𝐼2 | + 12|𝐸 ′ |,

𝑃∈𝑆

or equivalently, 2 3 |𝐼1 |

+ 13 |𝐼2 | + 12 |𝐸 ′ | ≤

𝑛 24

+ 14 .

(6.11)

Substituting Inequality (6.11) into Inequality (6.4) yields 𝛾(𝐺) ≤ |𝐷| ≤ 38 𝑛 + 14 . With additional work using similar techniques, Reed [655] showed that the 14 -term can be omitted in Inequality (6.11) to yield the desired bound 𝛾(𝐺) ≤ 38 𝑛. That the bound in Theorem 6.20 is tight may be seen as follows. Let F≥3 be the family of connected graphs with minimum degree at least 3 that can be obtained from a connected graph 𝐹 by adding a vertex-disjoint copy of the cubic graph 𝐻1 shown in Figure 6.9 for each vertex 𝑣 of 𝐹 and identifying the vertex 𝑣 of 𝐹 with the vertex 𝑐 0 of 𝐻1 . Let 𝐺 denote the resulting graph of order 𝑛 = 8|𝑉 (𝐹)| and minimum degree 3. In the special case when the graph 𝐹 is the cycle 𝐶4 , then the associated graph 𝐺 constructed from 𝐹 is illustrated in Figure 6.10. No two vertices in each copy of the graph 𝐻1 dominate all but the top, highlighted vertex in that copy. Every dominating set in 𝐺 therefore contains at least three vertices from each copy of the graph 𝐻1 , and so 𝛾(𝐺) ≥ 3|𝑉 (𝐹)| = 38 𝑛. By Theorem 6.20, 𝛾(𝐺) ≤ 38 𝑛. Consequently, 𝛾(𝐺) = 38 𝑛.

Figure 6.10 A graph in the family F≥3 In 2009 Shan et al. [673] modified the proof techniques employed by Reed [655] to obtain the following generalization of the result in Theorem 6.20 by allowing vertices of degree 2 back into the mix. Theorem 6.21 ([673]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and 𝑛2 vertices of degree 2, then 𝛾(𝐺) ≤ 38 𝑛 + 18 𝑛2 . If 𝑛2 = 0, then Theorem 6.21 reduces to Reed’s result given by Theorem 6.20. As an immediate consequence of Theorem 6.21, we have the following. Corollary 6.22 ([673]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and fewer than vertices of degree 2, then 𝛾(𝐺) < 25 𝑛.

𝑛 5

By Corollary 6.22, if 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and fewer than 𝑛5 vertices of degree 2, then the result of Theorem 6.21 is better than that of Theorem 6.18.

Section 6.2. Bounds on the Domination Number

149

As a special case of Theorem 6.20, we have the following result. Theorem 6.23 ([655]) If 𝐺 is a cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 83 𝑛. The two non-planar cubic graphs 𝐺 of order 𝑛 = 8 shown in Figure 6.11 satisfy 𝛾(𝐺) = 3 = 38 𝑛, showing that the upper bound in Theorem 6.23 is achievable.

(a)

(b)

Figure 6.11 The two non-planar cubic graphs of order 𝑛 = 8 We remark that Reed’s proof of Theorem 6.20 (and Theorem 6.23), which uses ingenious counting arguments, is technical in parts. Indeed some cases in the proof (namely, Claims 6.20.3 and 6.20.4) are best checked using a computer. In 2015 Dorbec et al. [236] presented a completely different proof to establish the 38 -bound on the domination number of a cubic graph that does not need a computer to check. A subcubic graph is a graph with maximum degree at most 3. Theorem 6.24 ([236]) If 𝐺 is a subcubic graph with 𝑛𝑖 vertices of degree 𝑖 for 𝑖 ∈ [3] 0 , then 8𝛾(𝐺) ≤ 8𝑛0 + 5𝑛1 + 4𝑛2 + 3𝑛3 . As a special case in Theorem 6.24 when 𝐺 is a cubic graph, we have the important result given in Theorem 6.23, noting that in this case 𝑛0 = 𝑛1 = 𝑛2 = 0. If 𝐺 is a subcubic graph of order 𝑛 and size 𝑚 with 𝑖 isolated vertices, then we note that 8𝑛0 + 5𝑛1 + 4𝑛2 + 3𝑛3 = 6𝑛 − 2𝑚 + 2𝑖. Hence, as an immediate consequence of Theorem 6.24, we have the following result from 1999 due to Fisher et al. [303] and Rautenbach [649]. Corollary 6.25 ([303, 649]) If 𝐺 is a subcubic graph of order 𝑛 and size 𝑚 with 𝑖 isolated vertices, then 4𝛾(𝐺) ≤ 3𝑛 − 𝑚 + 𝑖. As observed earlier, the two non-planar cubic graphs 𝐺 of order 𝑛 = 8 shown in Figure 6.11 satisfy 𝛾(𝐺) = 3 = 38 𝑛. In 2009 Kostochka and Stocker [537] proved that these two non-planar cubic graphs are the only connected cubic graphs that achieve the three-eights bound in Theorem 6.23. Excluding these graphs gave the following improved bound. 5 Theorem 6.26 ([537]) If 𝐺 is a connected cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 14 𝑛, unless 𝐺 is one of the two non-planar cubic graphs of order 𝑛 = 8 (shown in Figu1 re 6.11) in which case 𝛾(𝐺) = 3 = 14 (5𝑛 + 2).

For 𝑝 ≥ 3, 𝑝 > 𝑘 ≥ 1, and gcd( 𝑝, 𝑘) = 1, a generalized Petersen graph 𝑃( 𝑝, 𝑘) is the graph with vertex set {𝑣 0 , 𝑣 1 , . . . , 𝑣 𝑝−1 } ∪ {𝑤 0 , 𝑤 1 , . . . , 𝑤 𝑝−1 } and edges 𝑣 𝑖 𝑤 𝑖 ,

150

Chapter 6. Upper Bounds in Terms of Minimum Degree

𝑣 𝑖 𝑣 𝑖+1 and 𝑤 𝑖 𝑤 𝑖+𝑘 for 𝑖 ∈ [ 𝑝 − 1] 0 and the subscript sum is taken modulo 𝑝. In the special case when 𝑝 = 5 and 𝑘 = 2, the graph 𝑃( 𝑝, 𝑘) is the famous Petersen graph, illustrated in Figure 6.12.

Figure 6.12 The Petersen graph 𝑃(5, 2) If 𝐺 is the generalized Petersen graph 𝑃(7, 2) shown in Figure 6.13, then 𝐺 5 is a connected cubic graph of order 𝑛 = 14 satisfying 𝛾(𝐺) = 5 = 14 𝑛. Hence, 5 the 14 -upper bound of Theorem 6.26 is achievable. More generally, Kostochka and 5  Stocker [537] showed that the upper bound 14 𝑛 on the domination number of a connected cubic graph of order 𝑛 is achievable for 𝑛 ∈ {10, 12, 14, 16, 18}.

Figure 6.13 The generalized Petersen graph 𝑃(7, 2) In 1996 Reed [655] conjectured that his upper bound given in Theorem   6.23 on the domination number of a cubic graph can be improved to 𝛾(𝐺) ≤ 13 𝑛 . In 2005 Kostochka and Stodolsky [538] disproved his conjecture by constructing a connected cubic graph 𝐺 on 60 vertices with 𝛾(𝐺) = 21 and presented a sequence {𝐺 𝑘 }∞ 𝑘=1 of connected cubic graphs with lim

𝑘→∞

𝛾(𝐺 𝑘 ) 8 1 1 ≥ = + . |𝑉 (𝐺 𝑘 )| 23 3 69

In 2006 Kelmans [521] constructed a connected cubic graph 𝐺 on 54 vertices with 𝛾(𝐺) = 19 and gave an infinite series of 2-connected cubic graphs 𝐻 𝑘 with lim

𝑘→∞

𝛾(𝐻 𝑘 ) 1 1 ≥ + . |𝑉 (𝐻 𝑘 )| 3 60

Section 6.2. Bounds on the Domination Number

151

𝑛 denotes the family of all connected cubic graphs of order 𝑛, then as a If Gcubic consequence of Theorem 6.26 and the above result due to Kelmans,   1 1 𝛾(𝐺) 5 1 1 0.35 = + ≤ lim sup ≤ = + ≈ 0.3571. 𝑛 3 60 𝑛→∞ 𝐺 ∈ Gcubic 𝑛 14 3 42

It remains, however, an open problem to determine the limit of the supremum of the ratio of the domination number over the order of a connected cubic graph. It is not 5 known if there are graphs of large order that achieve the 14 -bound in Theorem 6.26. The problem of determining a tight upper bound on the domination number of a connected cubic graph of sufficiently large order, in terms of its order, remains one of the major outstanding problems in domination theory.

6.2.4

Minimum Degree Four

We remark that the results of Ore given in Theorem 6.2, Blank [79] and McCuaig and Shepherd [586] given in Theorem 6.18, and Reed [655] given in Theorem 6.20, all have the same form which we present in Theorem 6.27. Theorem 6.27 If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) = 𝛿 where 𝛿 ∈ [3], then   𝛿 𝛾(𝐺) ≤ 𝑛, 3𝛿 − 1 unless 𝛿 = 2 and 𝐺 is one of the seven exceptional graphs in the family Bdom shown in Figure 6.1. Motivated by the form of the bound on the domination number given in Theorem 6.27, Haynes et al. [417] conjectured in 1998 that this upper bound on the domination number holds for any minimum degree 𝛿 ≥ 1. Conjecture 6.28 ([417]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) = 𝛿 ≥ 1, then   𝛿 𝛾(𝐺) ≤ 𝑛, 3𝛿 − 1 unless 𝛿 = 2 and 𝐺 is one of the seven graphs in the family Bdom shown in Figure 6.1. Conjecture 6.28 is true as the following results, namely Theorems 6.29 and 6.35, show. In fact, we will give an improved bound in Chapter 7. In 2009 Sohn and Xudong [682] proved that Conjecture 6.28 holds for 𝛿 = 4, that 4 is, they proved that the 38 -bound in Theorem 6.20 can be improved to a 11 -bound if the minimum degree is at least 4. Theorem 6.29 ([682]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 4, then 𝛾(𝐺) ≤

4 11 𝑛.

The proof of Sohn and Xudong [682] uses the vertex-disjoint path cover method first employed by Reed [655], as demonstrated in the proof of Theorem 6.20.

152

Chapter 6. Upper Bounds in Terms of Minimum Degree

4 In 2021 Bujtás [123] gave a simpler proof of the 11 -bound on the domination number of graph with minimum degree 4 that combines an algorithmic approach using weighting arguments together with discharging methods. In order to present her proof, we first define the concept of a residual graph.

Definition 6.30 Given a graph 𝐺 with vertex set 𝑉 = 𝑉 (𝐺) and a subset 𝐷 ⊆ 𝑉, the residual graph denoted by 𝐺 𝐷 is the graph obtained from 𝐺 by assigning colors to the vertices and deleting certain edges according to the following rules: (a) A vertex 𝑣 is white if 𝑣 ∉ N[𝐷]. (b) A vertex 𝑣 is blue if 𝑣 ∈ N[𝐷] and N[𝑣] ⊈ N[𝐷]. (c) A vertex 𝑣 is red if N[𝑣] ⊆ N[𝐷]. (d) The edges of 𝐺 𝐷 consist of all edges of 𝐺 that are incident with at least one white vertex. Thus, in a residual graph 𝐺 𝐷 associated with a set 𝐷 of vertices of 𝐺, a vertex not dominated by 𝐷 is colored white, a vertex dominated by 𝐷 but with at least one neighbor not yet dominated by 𝐷 (and therefore colored white) is colored blue, and a vertex whose closed neighborhood is dominated by 𝐷 is colored red. We denote the set of white, blue, and red vertices by 𝑊, 𝐵, and 𝑅, respectively. We note that 𝐷 ⊆ 𝑅 and 𝑉 = 𝑊 ∪ 𝐵 ∪ 𝑅. As an illustration, consider the graph 𝐺 and the set 𝐷 of vertices shown in Figure 6.14(a), where the vertices in 𝐷 are highlighted. Applying Bujtás’s coloring rules, we color the vertices of 𝐺 with the colors white, blue, and red as shown in Figure 6.14(b). Thereafter, we delete all edges that are not incident with at least one white vertex from the graph, that is, we remove from the graph all edges incident with two red vertices or with two blue vertices or with one red vertex and one blue vertex. The resulting graph is the residual graph 𝐺 𝐷 shown in Figure 6.14(c).

(a) A set 𝐷 of vertices in a graph 𝐺

(b) The associated coloring of the vertices in 𝐺

(c) The residual graph 𝐺 𝐷

Figure 6.14 The residual graph 𝐺 𝐷 of a graph 𝐺 with a given set 𝐷 ⊂ 𝑉 (𝐺) We define the white-degree of a vertex 𝑣 in 𝐺 𝐷 , denoted by deg𝑊 (𝑣), as the number of white neighbors of the vertex 𝑣, and we define the blue-degree of a (white) vertex 𝑣 in 𝐺 𝐷 , denoted by deg 𝐵 (𝑣), as the number of blue neighbors of the vertex 𝑣.

Section 6.2. Bounds on the Domination Number

153

The maximum white-degree over all white and blue vertices, respectively, we denote by Δ𝑊 (𝑊) and Δ𝑊 (𝐵), respectively. By Definition 6.30, we have the following observation. Observation 6.31 ([123]) The following hold in the residual graph 𝐺 𝐷 : (a) Every white vertex 𝑣 has no red neighbor, and so deg𝐺 (𝑣) = deg𝑊 (𝑣) + deg 𝐵 (𝑣) ≥ 𝛿(𝐺). (b) A blue vertex 𝑣 has at least one white neighbor, and no blue or red neighbors. Further, if 𝑣 is a blue vertex, then at least one neighbor of 𝑣 in 𝐺 is colored red or blue in 𝐺 𝐷 , implying that deg𝑊 (𝑣) < deg𝐺 (𝑣). (c) Every red vertex is isolated in 𝐺 𝐷 . (d) If 𝐷 ⊂ 𝐷 ′ ⊆ 𝑉, then every red vertex in 𝐺 𝐷 remains red in 𝐺 𝐷 ′ , while every blue vertex in 𝐺 𝐷 is colored blue or red in 𝐺 𝐷 ′ . The number of white neighbors of a vertex in 𝐺 𝐷 ′ is therefore at most the number of its white neighbors in 𝐺 𝐷 . (e) The set 𝐷 is a dominating set in 𝐺 if and only if every vertex of 𝐺 𝐷 is colored red, that is, 𝑅 = 𝑉. 4 We now present the simpler proof due to Bujtás [123] of the 11 -bound on the domination number given in Theorem 6.29, which we restate here.

Theorem 6.29 ([682]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 4, then 𝛾(𝐺) ≤

4 11 𝑛.

Proof Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺) ≥ 4 and let 𝐷 ⊆ 𝑉. We define 𝐵4 as the set of blue vertices of degree at least 4 in 𝐺 𝐷 , and we define 𝐵𝑖 as the set of blue vertices of degree exactly 𝑖 in 𝐺 𝐷 for 𝑖 ∈ [3]. Thus, every (blue) vertex in 𝐵4 has at least four white neighbors, while every (blue) vertex in 𝐵𝑖 has exactly 𝑖 white neighbors for 𝑖 ∈ [3]. A vertex is a blue leaf if it belongs to 𝐵1 . For a vertex 𝑣 in the residual graph 𝐺 𝐷 , we define its weight w(𝑣) by the values given in Table 6.1.

set containing 𝑣

𝑊

𝐵4

𝐵3

𝐵2

𝐵1

𝑅

w(𝑣)

16

10

9

8

7

0

Table 6.1 The weight w(𝑣) assigned to a vertex 𝑣 in the residual graph 𝐺 𝐷

We define the weight of the residual graph 𝐺 𝐷 as ∑︁ w(𝐺 𝐷 ) = w(𝑣) = 16|𝑊 | + 10|𝐵4 | + 9|𝐵3 | + 8|𝐵2 | + 7|𝐵1 |. 𝑣 ∈𝑉

By Observation 6.31(e), w(𝐺 𝐷 ) = 0 if and only if 𝐷 is a dominating set in 𝐺. Given the graph 𝐺 and the set 𝐷, and given a subset 𝐴 ⊆ 𝑉 \ 𝐷, we define 𝜓( 𝐴) = w(𝐺 𝐷 ) − w(𝐺 𝐴∪𝐷 )

154

Chapter 6. Upper Bounds in Terms of Minimum Degree

as the decrease in weight when extending the set 𝐷 to the set 𝐴 ∪ 𝐷. We define a non-empty set 𝐴 ⊆ 𝑉 \ 𝐷 to be a 𝐷-desirable set if 𝜓( 𝐴) ≥ 44| 𝐴|. For 𝐴 ⊆ 𝑉 \ 𝐷, we define 𝐷 ′ = 𝐴 ∪ 𝐷, and we define 𝑊 ′ , 𝐵′ and 𝑅 ′ as the set of white, blue and red vertices, respectively, in 𝐺 𝐷 ′ . We define 𝐵4′ as the set of blue vertices of degree at least 4 in 𝐺 𝐷 ′ , and 𝐵𝑖′ as the set of blue vertices of degree exactly 𝑖 in 𝐺 𝐷 ′ for 𝑖 ∈ [3]. We proceed further by proving that the following property holds. Claim 6.29.1 If w(𝐺 𝐷 ) > 0, then the graph 𝐺 contains a 𝐷-desirable set. Proof Suppose, to the contrary, that w(𝐺 𝐷 ) > 0 but the graph 𝐺 does not contain a 𝐷-desirable set. In what follows, we present a series of subclaims describing some structural properties of 𝐺, which culminate in the existence of a 𝐷-desirable set. Subclaim 6.29.1.1 Δ𝑊 (𝑊) ≤ 2 and Δ𝑊 (𝐵) ≤ 3. Proof Suppose that Δ𝑊 (𝑊) ≥ 5. In this case, let 𝑣 be a white vertex for which deg𝑊 (𝑣) = Δ𝑊 (𝑊), and let 𝐴 = {𝑣}. In the residual graph 𝐺 𝐷 ′ , vertex 𝑣 is recolored red and its white neighbors are recolored blue or red. Thus, the weight of 𝑣 decreases by 16, while the weight of each white neighbor of 𝑣 decreases by at least 16 − 10 = 6. Thus, 𝜓( 𝐴) ≥ 16 + 5 · 6 = 46 > 44 = 44| 𝐴|, and so the set 𝐴 is a 𝐷-desirable set, contradicting our supposition that no such set exists. Hence, Δ𝑊 (𝑊) ≤ 4. Suppose that Δ𝑊 (𝑊) = 4. As before, let 𝑣 be a white vertex for which deg𝑊 (𝑣) = Δ𝑊 (𝑊), and let 𝐴 = {𝑣}. In the residual graph 𝐺 𝐷 ′ , vertex 𝑣 is recolored red and its white neighbors are recolored blue or red. Let 𝑢 be an arbitrary white neighbor of 𝑣 in 𝐺 𝐷 . The vertex 𝑢 has at most four white neighbors in 𝐺 𝐷 , and therefore has at most three white neighbors in 𝐺 𝐷 ′ . Thus, 𝑢 ∈ 𝐵3′ ∪ 𝐵2′ ∪ 𝐵1′ ∪ 𝑅 ′ in 𝐺 𝐷 ′ , implying that its weight decreases by at least 16 − 9 = 7. Therefore, 𝜓( 𝐴) ≥ 16 + 4 · 7 = 44 = 44| 𝐴|, a contradiction. Hence, Δ𝑊 (𝑊) ≤ 3. Suppose that Δ𝑊 (𝐵) ≥ 5. Let 𝑣 be a blue vertex for which deg𝑊 (𝑣) = Δ𝑊 (𝐵), and let 𝐴 = {𝑣}. We note that 𝑣 ∈ 𝐵4 and 𝑣 ∈ 𝑅 ′ , and so the weight decrease of vertex 𝑣 is 10. Let 𝑢 be an arbitrary (white) neighbor of 𝑣 in 𝐺 𝐷 . Since Δ𝑊 (𝑊) ≤ 3, vertex 𝑢 has at most three white neighbors in 𝐺 𝐷 . Thus, 𝑢 ∈ 𝐵3′ ∪ 𝐵2′ ∪ 𝐵1′ ∪ 𝑅 ′ in 𝐺 𝐷 ′ , implying that its weight decreases by at least 16 − 9 = 7. Therefore, 𝜓( 𝐴) ≥ 10 + 5 · 7 = 45 > 44| 𝐴|, a contradiction. Hence, Δ𝑊 (𝐵) ≤ 4. Suppose that Δ𝑊 (𝑊) = 3. Since 𝛿(𝐺) ≥ 4, it follows from Observation 6.31(a) and (d) that every white vertex has at least one blue neighbor. Let 𝑣 be a white vertex for which deg𝑊 (𝑣) = Δ𝑊 (𝑊), and let 𝐴 = {𝑣}. In the residual graph 𝐺 𝐷 ′ , vertex 𝑣 is recolored red and its white neighbors are recolored blue or red. Let 𝑢 be an arbitrary white neighbor of 𝑣 in 𝐺 𝐷 . The vertex 𝑢 has at most two white neighbors in 𝐺 𝐷 ′ . Thus, 𝑢 ∈ 𝐵2′ ∪ 𝐵1′ ∪ 𝑅 ′ in 𝐺 𝐷 ′ , implying that its weight decreases by at least 16 − 8 = 8. Let 𝑁 𝑣 be the set consisting of 𝑣 and its three white neighbors in 𝐺 𝐷 . By our earlier observations, every vertex in 𝑁 𝑣 has a blue neighbor. Thus, there are ℓ ≥ 4 edges joining vertices in 𝑁 𝑣 to blue vertices. Since Δ𝑊 (𝐵) ≤ 4, these ℓ edges

Section 6.2. Bounds on the Domination Number

155

result in a decrease in the weight of contribution of the blue neighbors of vertices in 𝑁 𝑣 by at least ℓ ≥ 4. Therefore, 𝜓( 𝐴) ≥ 16 + 3 · 8 + 4 = 44| 𝐴|, a contradiction. Hence, Δ𝑊 (𝑊) ≤ 2. Suppose that Δ𝑊 (𝐵) = 4. Let 𝑣 be a blue vertex for which deg𝑊 (𝑣) = Δ𝑊 (𝐵), and let 𝐴 = {𝑣}. We note that 𝑣 ∈ 𝐵4 and 𝑣 ∈ 𝑅 ′ , and so the weight decrease of vertex 𝑣 is 10. Let 𝑢 be an arbitrary (white) neighbor of 𝑣 in 𝐺 𝐷 . Since Δ𝑊 (𝑊) ≤ 2, the vertex 𝑢 has at most two white neighbors in 𝐺 𝐷 and at least one blue neighbor different from 𝑣. Thus, 𝑢 ∈ 𝐵2′ ∪ 𝐵1′ ∪ 𝑅 ′ in 𝐺 𝐷 ′ , implying that its weight decreases by at least 16 − 8 = 8. Further, there are ℓ ≥ 4 edges joining white neighbors of 𝑣 in 𝐺 𝐷 to blue vertices. These ℓ edges result in a decrease in the weight contribution of the blue neighbors of vertices in 𝑁 𝑣 by at least ℓ ≥ 4. Therefore, 𝜓( 𝐴) ≥ 10 + 4 · 8 + 4 = 46 > 44| 𝐴|, a contradiction. Hence, Δ𝑊 (𝐵) ≤ 3. This completes the proof of Subclaim 6.29.1.1. By Subclaim 6.29.1.1, Δ𝑊 (𝑊) ≤ 2 and Δ𝑊 (𝐵) ≤ 3. Subclaim 6.29.1.2 Each component in the subgraph 𝐺 𝐷 [𝑊] of 𝐺 𝐷 induced by the set 𝑊 of white vertices is a path 𝑃1 or 𝑃2 , or a cycle 𝐶4 or 𝐶7 . Proof Since Δ𝑊 (𝑊) ≤ 2, every component in the subgraph 𝐺 𝐷 [𝑊] of 𝐺 𝐷 induced by the set 𝑊 of white vertices is a path or a cycle. Suppose that 𝐺 𝐷 [𝑊] contains a path component 𝑣 1 𝑣 2 . . . 𝑣 𝑗 where 𝑗 ≥ 3. We let 𝐴 = {𝑣 2 }. In the graph 𝐺 𝐷 ′ , both 𝑣 1 and 𝑣 2 belong to 𝑅 ′ , and so each has a weight decrease of 16. The vertex 𝑣 3 belongs to the set 𝐵1′ ∪ 𝑅 ′ in 𝐺 𝐷 ′ . If 𝑣 3 ∈ 𝐵1′ , then its weight decrease is 16 − 7 = 9, while if 𝑣 3 ∈ 𝑅 ′ , then its weight decrease is 16. Let 𝑋 = {𝑣 1 , 𝑣 2 , 𝑣 3 } and note that there are at least 3 + 2 + 2 = 7 edges that join vertices in 𝑋 in 𝐺 𝐷 to blue vertices. These edges result in a decrease in the weight of contribution of the blue neighbors of vertices in 𝑋 by at least 7. Therefore, 𝜓( 𝐴) ≥ 2 · 16 + 9 + 7 = 48 > 44| 𝐴|, a contradiction. Hence, every path component in 𝐺 𝐷 [𝑊] is either a path 𝑃1 or 𝑃2 . Suppose next that 𝐺 𝐷 [𝑊] contains a cycle component 𝐶 : 𝑣 1 𝑣 2 . . . 𝑣 𝑞 for some 𝑞 ≥ 3. In this case, we let 𝐴 = {𝑣 𝑞 } ∪

⌊𝑞/3⌋ Ø

! {𝑣 3𝑖 } .

𝑖=1

Suppose firstly that 𝑞 ≡ 0 (mod 3). Thus, 𝑞 = 3𝑘 for some 𝑘 ≥ 1. In this case, 𝐴 = {𝑣 3 , . . . , 𝑣 3𝑘 } and | 𝐴| = 𝑘. In the graph 𝐺 𝐷 ′ , all vertices of 𝐶 belong to 𝑅 ′ , and therefore result in a decrease in the weight by 16 · 3𝑘 = 48𝑘. We note that there are at least 2 · 3𝑘 = 6𝑘 edges that join vertices in 𝑉 (𝐶) in 𝐺 𝐷 to blue vertices. These edges result in a decrease in the weight of contribution of the blue neighbors of vertices in 𝑉 (𝐶) by at least 6𝑘. Therefore, 𝜓( 𝐴) ≥ 48𝑘 + 6𝑘 = 54𝑘 > 44𝑘 = 44| 𝐴|, a contradiction. Suppose secondly that 𝑞 ≡ 2 (mod 3). Thus, 𝑞 = 3𝑘 + 2 for some 𝑘 ≥ 1. In this case, 𝐴 = {𝑣 3 , . . . , 𝑣 3𝑘 , 𝑣 3𝑘+2 } and | 𝐴| = 𝑘 + 1. Using similar arguments as before, we have 𝜓( 𝐴) ≥ 16 · (3𝑘 + 2) + 2 · (3𝑘 + 2) = 54𝑘 + 36 ≥ 44𝑘 + 44 = 44| 𝐴|, a contradiction.

156

Chapter 6. Upper Bounds in Terms of Minimum Degree

Suppose thirdly that 𝑞 ≡ 1 (mod 3). Thus, 𝑞 = 3𝑘 + 1 for some 𝑘 ≥ 1. In this case, 𝐴 = {𝑣 3 , . . . , 𝑣 3𝑘 , 𝑣 3𝑘+1 } and | 𝐴| = 𝑘 + 1. Using similar arguments as before, we have 𝜓( 𝐴) ≥ 16 · (3𝑘 + 1) + 2 · (3𝑘 + 1) = 54𝑘 + 18. If 𝑘 ≥ 3, then 𝜓( 𝐴) ≥ 54𝑘 + 18 > 44𝑘 + 44 = 44| 𝐴|, a contradiction. Hence, 𝑘 ∈ [2], that is, the component 𝐶 is either 𝐶4 or 𝐶7 . This completes the proof of Subclaim 6.29.1.2. Let 𝑊𝑖 be the set of white vertices having exactly 𝑖 white neighbors in 𝐺 𝐷 for 𝑖 ∈ [2] 0 . We note that 𝑊0 consists of all vertices of 𝑊 that belong to 𝑃1 -components of 𝐺 𝐷 [𝑊], while 𝑊1 consists of all vertices of 𝑊 that belong to 𝑃2 -components of 𝐺 𝐷 [𝑊]. By Subclaim 6.29.1.2, the set 𝑊2 consists of all vertices of 𝑊 that belong to a 𝐶4 -component or a 𝐶7 -component of 𝐺 𝐷 [𝑊]. Subclaim 6.29.1.3 No vertex of 𝐵3 is adjacent to a vertex of 𝑊0 in 𝐺 𝐷 . Proof Suppose, to the contrary, that there is a vertex 𝑣 ∈ 𝐵3 that has a neighbor 𝑢 ∈ 𝑊0 . Let 𝑢 1 and 𝑢 2 denote the other two white neighbors of 𝑣 in 𝐺 𝐷 , and let 𝐴 = {𝑣}. In the graph 𝐺 𝐷 ′ we note that 𝑣 ∈ 𝑅 ′ and 𝑢 ∈ 𝑅 ′ , and so the weight decrease of the vertices 𝑣 and 𝑢 are 9 and 16, respectively. The vertices 𝑢 1 and 𝑢 2 belong to the set 𝐵1′ ∪ 𝐵2′ ∪ 𝑅 ′ in 𝐺 𝐷 ′ , and therefore have a combined weight decrease of at least 2 · (16 − 8) = 16. Moreover, there are at least three edges that join 𝑢 to blue vertices different from 𝑣, and at least one edge that joins each of 𝑢 1 and 𝑢 2 to a blue vertex different from 𝑣. Thus, there are at least five edges that join vertices in {𝑢, 𝑢 1 , 𝑢 2 } to blue vertices different from 𝑣. These edges result in a decrease in the total weight of at least 5, implying that 𝜓( 𝐴) ≥ 9 + 16 + 16 + 5 = 46 > 44𝑘 = 44| 𝐴|, a contradiction. By Subclaim 6.29.1.3, every neighbor of a vertex in 𝑊0 belongs to the set 𝐵1 ∪ 𝐵2 in the graph 𝐺 𝐷 . Adopting the notation of Bujtás [123], we call a vertex special if it belongs to 𝐵2 and is adjacent to a vertex of 𝑊0 in 𝐺 𝐷 . Subclaim 6.29.1.4 The following hold: (a) No special vertex is adjacent to two vertices of 𝑊0 in 𝐺 𝐷 . (b) No special vertex is adjacent to a vertex of 𝑊2 in 𝐺 𝐷 . (c) If 𝑣 1 and 𝑣 2 are two adjacent vertices in 𝑊1 , then at most one of them has a special (blue) neighbor. Proof (a) Suppose, to the contrary, that a special vertex 𝑣 ∈ 𝐵2 has two neighbors 𝑢 1 and 𝑢 2 that belong to the set 𝑊0 . We let 𝐴 = {𝑣}. In the graph 𝐺 𝐷 ′ we note that {𝑢 1 , 𝑢 2 , 𝑣} ⊆ 𝑅 ′ , and so the weight decrease of vertex 𝑣 is 8, while the weight decrease of each of 𝑢 1 and 𝑢 2 is 16. There are at least six edges that join vertices in {𝑢 1 , 𝑢 2 } to blue vertices different from 𝑣. These edges result in a decrease in the total weight of at least 6, implying that 𝜓( 𝐴) ≥ 8 + 2 · 16 + 6 = 46 > 44𝑘 = 44| 𝐴|, a contradiction. This proves part (a). (b) Suppose, to the contrary, that a special vertex 𝑣 ∈ 𝐵2 has a neighbor 𝑢 0 that belongs to 𝑊0 and a neighbor 𝑢 1 that belongs to 𝑊2 . By our earlier observations, 𝑢 1 belongs to a 𝐶4 -component or a 𝐶7 -component 𝐶 of 𝐺 𝐷 [𝑊]. Suppose firstly that 𝐶 is a 4-cycle given by 𝐶 : 𝑢 1 𝑢 2 𝑢 3 𝑢 4 𝑢 1 . In this case, we let 𝐴 = {𝑣, 𝑢 3 }. In the graph 𝐺 𝐷 ′ we note that {𝑣, 𝑢 0 , 𝑢 1 , 𝑢 2 , 𝑢 3 , 𝑢 4 } ⊆ 𝑅 ′ , and so the weight decrease of vertex 𝑣 is 8, while the weight decrease of each vertex in

Section 6.2. Bounds on the Domination Number

157

𝑉 (𝐶) ∪ {𝑢 0 } is 16. There are at least three edges that join 𝑢 0 to blue vertices different from 𝑣, at least one edge that joins 𝑢 1 to a blue vertex different from 𝑣, and at least two edges that join 𝑢 𝑖 to a blue vertex different from 𝑣 for 𝑖 ∈ [4] \ {1}. Thus, there are at least ten edges that join vertices in {𝑢 0 , 𝑢 1 , 𝑢 2 , 𝑢 3 , 𝑢 4 } to blue vertices different from 𝑣. These edges result in a decrease in the total weight of at least 10, implying that 𝜓( 𝐴) ≥ 8 + 5 · 16 + 10 = 98 > 44| 𝐴|, a contradiction. Suppose secondly that 𝐶 is a 7-cycle given by 𝐶 : 𝑢 1 𝑢 2 . . . 𝑢 7 𝑢 1 . In this case, we let 𝐴 = {𝑣, 𝑢 3 , 𝑢 6 }. Using similar arguments as before, we have 𝜓( 𝐴) ≥ 8+8·16+16 = 152 > 44| 𝐴|, a contradiction. This proves part (b). (c) Suppose, to the contrary, that there is a 𝑃2 -component of 𝐺 𝐷 [𝑊] in which both vertices 𝑣 1 and 𝑣 2 have special (blue) neighbors. Let 𝑢 𝑖 be a special (blue) neighbor of 𝑣 𝑖 and let 𝑥𝑖 be the second white neighbor of 𝑢 𝑖 for 𝑖 ∈ [2]. We note that {𝑥 1 , 𝑥2 } ⊆ 𝑊0 , {𝑣 1 , 𝑣 2 } ⊆ 𝑊1 , {𝑢 1 , 𝑢 2 } ⊆ 𝐵2 , and 𝑢 1 ≠ 𝑢 2 . We now let 𝐴 = {𝑢 1 , 𝑢 2 }. In the graph 𝐺 𝐷 ′ we note that {𝑢 1 , 𝑢 2 , 𝑣 1 , 𝑣 2 , 𝑥1 , 𝑥2 } ⊆ 𝑅 ′ , and so the weight decrease of each vertex in {𝑢 1 , 𝑢 2 } is 8, while the weight decrease of each vertex in {𝑣 1 , 𝑣 2 , 𝑥1 , 𝑥2 } is 16. There are at least six edges that join vertices in {𝑥 1 , 𝑥2 } to blue vertices different from 𝑢 1 and 𝑢 2 , and there are at least four edges that join vertices in {𝑣 1 , 𝑣 2 } to blue vertices different from 𝑢 1 and 𝑢 2 . Thus, there are at least ten edges that join vertices in {𝑣 1 , 𝑣 2 , 𝑥1 , 𝑥2 } to blue vertices different from 𝑢 1 and 𝑢 2 . These edges result in a decrease in the total weight of at least 10, implying that 𝜓( 𝐴) ≥ 2 · 8 + 4 · 16 + 10 = 90 > 44| 𝐴|, a contradiction. This proves part (c), and completes the proof of Subclaim 6.29.1.4. We now return to the proof of Claim 6.29.1. With the structure of the residual graph 𝐺 𝐷 given by Subclaims 6.29.1.1, 6.29.1.2, 6.29.1.3, and 6.29.1.4, we apply the following discharging arguments that distribute the weights of the vertices. Discharging Rules. We initially assign charges to the (non-red) vertices of 𝐺 𝐷 so that every white vertex receives a weight of 16, and every (blue) vertex in 𝐵3 , 𝐵2 , and 𝐵1 receives a weight of 9, 8, and 7, respectively. Since Δ𝑊 (𝐵) ≤ 3, we note that the sum of the charges equals w(𝐺 𝐷 ). Thereafter, we distribute the charge of every blue vertex that is not a special vertex equally amongst its white neighbors, while we distribute the charge of a special vertex unequally amongst its two neighbors by giving a charge of 7 to its (unique) neighbor in 𝑊0 and a charge of 1 to its other neighbor. Thus, • Every vertex of 𝐵3 gives 3 to each of its three white neighbors. • Every non-special vertex of 𝐵2 gives 4 to both of its white neighbors. • Every vertex of 𝐵1 gives 7 to its (unique) white neighbor. • Every special vertex of 𝐵2 gives 7 to its white neighbor in 𝑊0 and gives 1 to its other white neighbor. Subclaim 6.29.1.5 The following hold in the graph 𝐺 𝐷 [𝑊]: (a) Every 𝑃1 -component has a charge of at least 44. (b) Every 𝑃2 -component has a charge of at least 44. (c) Every 𝐶4 -component has a charge of at least 88. (d) Every 𝐶7 -component has a charge of at least 154.

158

Chapter 6. Upper Bounds in Terms of Minimum Degree

Proof Subclaim 6.29.1.3 implies that the open neighborhood of a vertex from a 𝑃1 -component of 𝐺 𝐷 [𝑊] is contained in 𝐵1 ∪ 𝐵2 . Thus, upon completion of the discharging process, every vertex from a 𝑃1 -component of 𝐺 𝐷 [𝑊] has a charge of at least 16 + 4 · 7 = 44. By Subclaim 6.29.1.4(c), if 𝑣 1 and 𝑣 2 are the two vertices of a 𝑃2 -component of 𝐺 𝐷 [𝑊], then at least one of 𝑣 1 and 𝑣 2 , say 𝑣 1 , has three non-special blue neighbors, and therefore receives a charge of at least 3 · 3 = 9, while 𝑣 2 receives a charge of at least 3 · 1. Hence, every 𝑃2 -component of 𝐺 𝐷 [𝑊] has a charge of at least 2 · 16 + 3 · 3 + 1 · 3 = 44. By Subclaim 6.29.1.4(b), no special vertex is adjacent to a vertex in a 𝐶4 - or 𝐶7 -component of 𝐺 𝐷 [𝑊]. Hence, if 𝐶 is a 𝐶4 -component of 𝐺 𝐷 [𝑊], then there are at least eight edges that join vertices in 𝑉 (𝐶) to blue vertices, implying that the vertices of 𝐶 receive a charge of at least 8 · 3 from blue neighbors of 𝐶. If 𝐶 is a 𝐶7 -component of 𝐺 𝐷 [𝑊], then there are at least fourteen edges that join vertices in 𝑉 (𝐶) to blue vertices, implying that the vertices of 𝐶 receive a charge of at least 14 · 3 from blue neighbors of 𝐶. Thus, every 𝐶4 -component of 𝐺 𝐷 [𝑊] has a charge of at least 4 · 16 + 8 · 3 = 88, while every 𝐶7 -component of 𝐺 𝐷 [𝑊] has a charge of at least 7 · 16 + 14 · 3 = 154. Let 𝑝 1 , 𝑝 2 , 𝑐 4 , and 𝑐 7 be the number of 𝑃1 -, 𝑃2 -, 𝐶4 -, and 𝐶7 -components of 𝐺 𝐷 [𝑊], respectively. Let 𝐴 be a 𝛾-set of 𝐺 [𝑊]. We note that | 𝐴| = 𝑝 1 + 𝑝 2 + 2𝑐 4 + 3𝑐 7 . The set 𝐷 ′ = 𝐴 ∪ 𝐷 is a dominating set of 𝐺 𝐷 ′ , and so w(𝐺 𝐷 ′ ) = 0. Thus, 𝜓( 𝐴) = w(𝐺 𝐷 ) − w(𝐺 𝐷 ′ ) = w(𝐺 𝐷 ). By the discharging rules defined earlier, the weight of the blue vertices in 𝐺 𝐷 is charged to the white vertices. Thus, by Subclaim 6.29.1.5, 𝜓( 𝐴) = w(𝐺 𝐷 ) ≥ 44𝑝 1 + 44𝑝 2 + 88𝑐 4 + 154𝑐 7 ≥ 44( 𝑝 1 + 𝑝 2 + 2𝑐 4 + 3𝑐 7 ) = 44| 𝐴|. The set 𝐴 is therefore a 𝐷-desirable set, contradicting our supposition that no such set exists. By Claim 6.29.1, if w(𝐺 𝐷 ) > 0, then the graph 𝐺 contains a 𝐷-desirable set. We now return to the proof of Theorem 6.29. Recall that 𝐺 is an arbitrary graph of order 𝑛 with 𝛿(𝐺) ≥ 4. Let 𝐷 0 = ∅ and let 𝐺 0 = 𝐺 𝐷0 . We note that 𝑉 (𝐺 0 ) = 𝑊 and w(𝐺 0 ) = 16𝑛. By Claim 6.29.1, there exists a 𝐷 0 -desirable set 𝐴1 . In this case, if we let 𝐷 1 = 𝐷 0 ∪ 𝐴1 = 𝐴1 (noting that 𝐷 0 = ∅) and 𝐺 1 = 𝐺 𝐷1 , then w(𝐺 0 ) − w(𝐺 1 ) ≥ 44| 𝐴1 |, that is,

Section 6.2. Bounds on the Domination Number

159

w(𝐺 1 ) ≤ w(𝐺 0 ) − 44| 𝐴1 |. If w(𝐺 1 ) > 0, then by Claim 6.29.1, there exists a 𝐷 1 -desirable set 𝐴2 . If we let 𝐷 2 = 𝐴1 ∪ 𝐴2 and 𝐺 2 = 𝐺 𝐷2 , then w(𝐺 1 ) − w(𝐺 2 ) ≥ 44| 𝐴2 |, that is, w(𝐺 2 ) ≤ w(𝐺 1 ) − 44| 𝐴2 |. If w(𝐺 2 ) > 0, then we continue this process. In this way, we obtain a sequence of residual graphs 𝐺 0 , 𝐺 1 , . . . , 𝐺 𝑘 and a dominating set 𝐷 = 𝐴1 ∪ · · · ∪ 𝐴 𝑘 of 𝐺 such that 0 = w(𝐺 𝑘 ) ≤ w(𝐺 𝑘−1 ) − 44| 𝐴 𝑘 | ≤ w(𝐺 0 ) + 44

𝑘 ∑︁

| 𝐴𝑖 |

𝑖=1

≤ 16𝑛 − 44

𝑘 ∑︁

| 𝐴𝑖 |

𝑖=1

= 16𝑛 − 44|𝐷|. Consequently, 𝛾(𝐺) ≤ |𝐷 | ≤

16 44 𝑛

=

4 11 𝑛.

At the 9th Cracow Conference on Graph Theory held in Ryto, Poland, in June 2022, 4 71 Bujtás improved the 11 -bound in Theorem 6.29 to essentially a 200 -bound. More precisely, she proved that if 𝐺 is a connected graph with minimum degree at least 4 71𝑛 of order 𝑛, then 𝛾(𝐺) ≤ 71𝑛+5 200 . Moreover, if 𝑛 is large enough, then 𝛾(𝐺) ≤ 200 . Henning [460] posed the following conjecture. Conjecture 6.32 ([460]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 4, then 𝛾(𝐺) ≤ 13 𝑛. If 𝐺 is the graph obtained from a complete graph 𝐾6 by removing the edges of a perfect matching, then 𝐺 is a 4-regular graph of order 𝑛 = 6 satisfying 𝛾(𝐺) = 2 = 13 𝑛. Hence, if Conjecture 6.32 is correct, then the bound is best possible.

6.2.5

Minimum Degree Five

In 2006 Xing et al. [754] proved that Conjecture 6.28 holds for 𝛿 = 5, that is, they 4 5 proved that the 11 -bound in Theorem 6.29 can be improved to a 14 -bound if we restrict the minimum degree to be at least 5. Theorem 6.33 ([754]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 5, then 𝛾(𝐺) ≤ 5 14 𝑛 < 0.3572 𝑛. The proof of Xing et al. [754] once again uses the vertex-disjoint path cover method first employed by Reed [655]. In 2016 Bujtás and Klavžar [125] improved this bound as follows.

Chapter 6. Upper Bounds in Terms of Minimum Degree

160

Theorem 6.34 ([125]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 5, then 𝛾(𝐺) ≤

2671 𝑛 < 0.3439 𝑛. 7766

In 2021 Bujtás [123] proved a breakthrough result that the domination number in graphs with minimum degree at least 5 is at most the magical threshold of 13 𝑛. The proof Bujtás presented of this result uses the same approach as her proof of Theorem 6.29 presented in the previous section, except that as expected in this case when the minimum degree is at least 5, the vertex weighting arguments, as well as the discharging methods employed, are more intricate and involved, and a more detailed case analysis is needed. We present here only an outline of her proof, where we adopt the notation defined in Section 6.2.4. Theorem 6.35 ([123]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 5, then 𝛾(𝐺) ≤ 13 𝑛. Proof Sketch Let 𝐺 = (𝑉, 𝐸) be a graph of order 𝑛 with 𝛿(𝐺) ≥ 5, and let 𝐷 ⊆ 𝑉. We define 𝐵5 as the set of blue vertices of degree at least 5 in 𝐺 𝐷 , and we define 𝐵𝑖 as the set of blue vertices of degree exactly 𝑖 in 𝐺 𝐷 for 𝑖 ∈ [4]. Thus, every (blue) vertex in 𝐵5 has at least five white neighbors, while every (blue) vertex in 𝐵𝑖 has exactly 𝑖 white neighbors for 𝑖 ∈ [4]. For a vertex 𝑣 in the residual graph 𝐺 𝐷 , we define its weight w(𝑣) by the values given in Table 6.2.

set containing 𝑣

𝑊

𝐵5

𝐵4

𝐵3

𝐵2

𝐵1

𝑅

w(𝑣)

35

23

21

19

17

14

0

Table 6.2 The weight w(𝑣) assigned to a vertex 𝑣 in the residual graph 𝐺 𝐷

We define the weight of the residual graph 𝐺 𝐷 as ∑︁ w(𝐺 𝐷 ) = w(𝑣) = 35|𝑊 | + 23|𝐵5 | + 21|𝐵4 | + 19|𝐵3 | + 17|𝐵2 | + 14|𝐵1 |. 𝑣 ∈𝑉

By Observation 6.31(e), w(𝐺 𝐷 ) = 0 if and only if 𝐷 is a dominating set in 𝐺. As in the proof of Theorem 6.29, given the graph 𝐺 and the set 𝐷, and given a subset 𝐴 ⊆ 𝑉 \ 𝐷, we define 𝜓( 𝐴) = w(𝐺 𝐷 ) − w(𝐺 𝐴∪𝐷 ) as the decrease in weight when extending the set 𝐷 to the set 𝐴 ∪ 𝐷. However, we now define a non-empty set 𝐴 ⊆ 𝑉 \ 𝐷 to be a 𝐷-desirable set if 𝜓( 𝐴) ≥ 105| 𝐴|. We proceed further by proving that the following property holds. Claim 6.35.1 If w(𝐺 𝐷 ) > 0, then the graph 𝐺 contains a 𝐷-desirable set. Proof Sketch Suppose, to the contrary, that w(𝐺 𝐷 ) > 0 but the graph 𝐺 does not contain a 𝐷-desirable set. As before, we let 𝑊𝑖 be the set of white vertices having

Section 6.2. Bounds on the Domination Number

161

exactly 𝑖 white neighbors in 𝐺 𝐷 for 𝑖 ∈ [2] 0 . We note that 𝑊0 consists of all vertices of 𝑊 that belong to 𝑃1 -components of 𝐺 𝐷 [𝑊]. We call a vertex special if it belongs to 𝐵2 and is adjacent to a vertex of 𝑊0 in 𝐺 𝐷 . In what follows, we present a series of subclaims (without proof) describing some structural properties of 𝐺, which show that 𝐺 has a 𝐷-desirable set. Subclaim 6.35.1.1 The following hold: (a) Δ𝑊 (𝑊) ≤ 2 and Δ𝑊 (𝐵) ≤ 3. (b) Each component in the subgraph 𝐺 𝐷 [𝑊] of 𝐺 𝐷 induced by the set 𝑊 of white vertices is a path 𝑃1 or 𝑃2 , or a cycle 𝐶4 , 𝐶5 , 𝐶7 , or 𝐶10 . (c) No vertex in 𝐵3 is adjacent to a vertex in 𝑊0 in 𝐺 𝐷 . (d) No special vertex is adjacent to two vertices in 𝑊0 in 𝐺 𝐷 . (e) No special vertex is adjacent to a vertex in a 𝐶4 - or 𝐶7 -component in 𝐺 𝐷 [𝑊]. (f) If 𝑣 1 and 𝑣 2 are two adjacent vertices in 𝑊1 , then at most one of them has a special (blue) neighbor. Discharging Rules. We initially assign charges to the (non-red) vertices of 𝐺 𝐷 so that every white vertex receives a weight of 35, and every (blue) vertex in 𝐵3 , 𝐵2 , and 𝐵1 receives a weight of 19, 17, and 14, respectively. Since Δ𝑊 (𝐵) ≤ 3, we note that the sum of the charges equals w(𝐺). Thereafter, we distribute the charge of every blue vertex that is not a special vertex equally amongst its white neighbors, while we distribute the charge of a special vertex unequally amongst its two neighbors by giving a charge of 14 to its (unique) neighbor in 𝑊0 and a charge of 3 to its other neighbor. Thus, • Every vertex in 𝐵3 gives 19 3 to each of its three white neighbors. • Every non-special vertex in 𝐵2 gives 17 2 to both of its white neighbors. • Every vertex in 𝐵1 gives 14 to its (unique) white neighbor. • Every special vertex in 𝐵2 gives 14 to its white neighbor in 𝑊0 and gives 3 to its other white neighbor. Upon completion of the discharging process, the charge of each white component is given as follows. Subclaim 6.35.1.2 The following hold in the graph 𝐺 𝐷 [𝑊]: (a) Every 𝑃1 -component has a charge of at least 105. (b) Every 𝑃2 -component has a charge of at least 107. (c) Every 𝐶4 -component has a charge of at least 216. (d) Every 𝐶5 -component has a charge of at least 220. (e) Every 𝐶7 -component has a charge of at least 378. (f) Every 𝐶10 -component has a charge of at least 440. Let 𝑝 1 , 𝑝 2 , 𝑐 4 , 𝑐 5 , 𝑐 7 , and 𝑐 10 be the number of 𝑃1 -, 𝑃2 -, 𝐶4 -, 𝐶5 -, 𝐶7 -, and 𝐶10 -components of 𝐺 𝐷 [𝑊], respectively. Let 𝐴 be a 𝛾-set of 𝐺 [𝑊]. We note that | 𝐴| = 𝑝 1 + 𝑝 2 + 2𝑐 4 + 2𝑐 5 + 3𝑐 7 + 4𝑐 10 . The set 𝐷 ′ = 𝐴 ∪ 𝐷 is a dominating set of 𝐺 𝐷 ′ , and so w(𝐺 𝐷 ′ ) = 0. Thus, 𝜓( 𝐴) = w(𝐺 𝐷 ) − w(𝐺 𝐷 ′ ) = w(𝐺 𝐷 ). By the discharging rules defined earlier,

Chapter 6. Upper Bounds in Terms of Minimum Degree

162

the weight of the blue vertices in 𝐺 𝐷 is charged to the white vertices. Thus, by Subclaim 6.35.1.2, 𝜓( 𝐴) = w(𝐺 𝐷 ) ≥ 105𝑝 1 + 107𝑝 2 + 216𝑐 4 + 220𝑐 5 + 378𝑐 7 + 440𝑐 10 ≥ 105( 𝑝 1 + 𝑝 2 + 2𝑐 4 + 2𝑐 5 + 3𝑐 7 + 4𝑐 10 ) = 105| 𝐴|. The set 𝐴 is therefore a 𝐷-desirable set, contradicting our supposition that no such set exists. By Claim 6.35.1, if w(𝐺 𝐷 ) > 0, then the graph 𝐺 contains a 𝐷-desirable set. We now return to the proof of Theorem 6.35. Recall that 𝐺 is an arbitrary graph of order 𝑛 with 𝛿(𝐺) ≥ 5. Let 𝐷 0 = ∅ and let 𝐺 0 = 𝐺 𝐷0 . We note that 𝑉 (𝐺 0 ) = 𝑊 and w(𝐺 0 ) = 35𝑛. By Claim 6.35.1, there exists a 𝐷 0 -desirable set 𝐴1 . If we let 𝐷 1 = 𝐷 0 ∪ 𝐴1 = 𝐴1 and 𝐺 1 = 𝐺 𝐷1 , then w(𝐺 0 ) − w(𝐺 1 ) ≥ 105| 𝐴1 |, that is, w(𝐺 1 ) ≤ w(𝐺 0 ) − 105| 𝐴1 |. If w(𝐺 1 ) > 0, then by Claim 6.35.1, there exists a 𝐷 1 -desirable set 𝐴2 . If we let 𝐷 2 = 𝐴1 ∪ 𝐴2 and 𝐺 2 = 𝐺 𝐷2 , then w(𝐺 1 ) − w(𝐺 2 ) ≥ 105| 𝐴2 |, that is, w(𝐺 2 ) ≤ w(𝐺 1 ) − 105| 𝐴2 |. If w(𝐺 2 ) > 0, then we continue this process. In this way, we obtain a sequence of residual graphs 𝐺 0 , 𝐺 1 , . . . , 𝐺 𝑘 and a dominating set 𝐷 = 𝐴1 ∪ · · · ∪ 𝐴 𝑘 of 𝐺 such that 0 = w(𝐺 𝑘 ) ≤ w(𝐺 𝑘−1 ) − 105| 𝐴 𝑘 | ≤ 35𝑛 − 105

𝑘 ∑︁

| 𝐴𝑖 | = 35𝑛 − 105|𝐷|.

𝑖=1

Consequently, 𝛾(𝐺) ≤ |𝐷| ≤

35 105 𝑛

= 13 𝑛.

The 13 -upper bound on the domination number of a graph with minimum degree at least 5 given in Theorem 6.35 is currently the best known bound. However, it is not known if this bound is achievable. We also note that Theorem 6.35 proves Conjecture 6.28 for minimum degree at least 5 and, combined with Theorems 6.27 and 6.29, proves the conjecture for all 𝛿 ≥ 1.

6.2.6

Minimum Degree Six

In 2016 Bujtás and Klavžar [125] established the best known upper bound up to that time on the domination number of a graph with minimum degree at least 6.

Section 6.2. Bounds on the Domination Number

163

Theorem 6.36 ([125]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then 𝛾(𝐺) ≤

1702 𝑛 ≈ 0.3158 𝑛. 5389

The upper bound given in Theorem 6.36 was subsequently improved in 2021 by Bujtás and Henning [124]. Their proof once again uses Bujtás approach of vertex weighting arguments and discharging methods, combined with a detailed case analysis. We present here a brief outline of the proof, which has the same flavor as the earlier proofs presented using this approach, but uses some new ideas and is more intricate and involved than the proofs of Theorems 6.29 and 6.35. Theorem 6.37 ([124]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then 𝛾(𝐺) ≤

127 𝑛 ≈ 0.3038 𝑛. 418

Proof Sketch Since removing edges from a graph cannot decrease its domination number, it suffices for us to prove the result for graphs 𝐺 = (𝑉, 𝐸) that are edge minimal with respect to the two conditions: (i) 𝐺 is connected and (ii) 𝛿(𝐺) ≥ 6. With this assumption, every edge is incident with at least one vertex of degree 6, and so the set of vertices of degree at least 7 in 𝐺 is an independent set. Let 𝐷 ⊆ 𝑉 (𝐺). We define 𝐵6 as the set of blue vertices of degree at least 6 in 𝐺 𝐷 , and we define 𝐵𝑖 as the set of blue vertices of degree exactly 𝑖 in 𝐺 𝐷 for 𝑖 ∈ [5]. For a vertex 𝑣 in the residual graph 𝐺 𝐷 , we define its weight w(𝑣) by the values given in Table 6.3.

set containing 𝑣

𝑊

𝐵6

𝐵5

𝐵4

𝐵3

𝐵2

𝐵1

𝑅

w(𝑣)

508

508

317.4

297

268

236

194

0

Table 6.3 The weight w(𝑣) assigned to a vertex 𝑣 in the residual graph 𝐺 𝐷

We define the weight of the residual graph 𝐺 𝐷 as ∑︁ w(𝐺 𝐷 ) = w(𝑣) = 508|𝑊 | + 508|𝐵6 | + 317.4|𝐵5 | 𝑣 ∈𝑉 (𝐺) + 297|𝐵4 | + 268|𝐵3 | + 236|𝐵2 | + 194|𝐵1 |. We observe that w(𝐺 𝐷 ) = 0 if and only if 𝐷 is a dominating set in 𝐺. As in the proof of Theorem 6.29, given the graph 𝐺 and the set 𝐷, and given a subset 𝐴 ⊆ 𝑉 \ 𝐷, we define 𝜓( 𝐴) = w(𝐺 𝐷 ) − w(𝐺 𝐴∪𝐷 ) as the decrease in weight when extending the set 𝐷 to the set 𝐴 ∪ 𝐷. However, we now define a non-empty set 𝐴 ⊆ 𝑉 \ 𝐷 to be a 𝐷-desirable set if 𝜓( 𝐴) ≥ 1672| 𝐴|. Further, we define 𝐷 ′ = 𝐴 ∪ 𝐷 and denote by 𝑊 ′ , 𝐵′ , and 𝑅 ′ the set of white, blue, and red vertices, respectively, in 𝐺 𝐷 ′ . We define 𝐵6′ as the set of blue vertices of

164

Chapter 6. Upper Bounds in Terms of Minimum Degree

degree at least 6 in 𝐺 𝐷 ′ , and 𝐵𝑖′ as the set of blue vertices of degree exactly 𝑖 in 𝐺 𝐷 ′ for 𝑖 ∈ [5]. A key property of the graph 𝐺 is given by the following claim. Claim 6.37.1 If w(𝐺 𝐷 ) > 0 and 𝐵6 = ∅ in 𝐺 𝐷 , then the graph 𝐺 contains a 𝐷-desirable set 𝐴 such that 𝐵6′ remains empty in 𝐺 𝐷 ′ . Proof Sketch Suppose, to the contrary, that w(𝐺 𝐷 ) > 0 and 𝐵6 = ∅, but the graph 𝐺 does not contain a 𝐷-desirable set with the given property. By supposition, 𝐵6 = ∅ in 𝐺 𝐷 , and so Δ𝑊 (𝐵) ≤ 5. In what follows, we present a series of subclaims (without proof) describing some structural properties of 𝐺, which prove that 𝐺 has a 𝐷-desirable set. Subclaim 6.37.1.1 The following hold: (a) Δ𝑊 (𝑊) ≤ 2 and Δ𝑊 (𝐵) ≤ 3. (b) Each component in the subgraph 𝐺 𝐷 [𝑊] of 𝐺 𝐷 induced by the set 𝑊 of white vertices is a path 𝑃1 or 𝑃2 , or a cycle 𝐶4 or 𝐶5 . Let 𝑊𝑖 be the set of white vertices having exactly 𝑖 white neighbors in 𝐺 𝐷 for 𝑖 ∈ [2] 0 . We call a vertex special if it belongs to 𝐵2 and is adjacent to a vertex of 𝑊0 in 𝐺 𝐷 . Subclaim 6.37.1.2 The following hold: (a) No vertex of 𝐵3 is adjacent to a vertex of 𝑊0 in 𝐺 𝐷 . (b) No special vertex is adjacent to two vertices of 𝑊0 in 𝐺 𝐷 . (c) No special vertex is adjacent to a vertex in a 𝐶4 -component in 𝐺 𝐷 [𝑊]. (d) If 𝑢 ∈ 𝑊0 and 𝑣 ∈ 𝑊1 , then there are at most two (special) vertices that have 𝑢 and 𝑣 as common neighbors in 𝐺 𝐷 . We next specify a set P ★ of 6-paths. A path 𝑥1 𝑥 2 . . . 𝑥6 of 𝐺 𝐷 is called a 𝑃★-path if 𝑥1 , 𝑥6 ∈ 𝑊0 , 𝑥2 , 𝑥5 ∈ 𝐵2 , and 𝑥 3 , 𝑥4 ∈ 𝑊1 . Let P ★ be a maximal set of pairwise vertex-disjoint 𝑃★-paths in 𝐺 𝐷 , where maximality means that every 𝑃★-path that does ★ not belong to P ★ shares Ð a vertex with at least one path in P . We say that a vertex 𝑣 is covered if 𝑣 ∈ 𝑃 ∈ P★ 𝑉 (𝑃). A special vertex 𝑣 ∈ 𝐵2 is called C-special and UC-special if its unique neighbor 𝑣 ′ in 𝑊0 is covered or uncovered, respectively. The maximum possible number of UC-special neighbors of an uncovered 𝑃2 -component of 𝐺 𝐷 [𝑊] is given by the following subclaim. Subclaim 6.37.1.3 If 𝑣 1 and 𝑣 2 form an uncovered 𝑃2 -component in 𝐺 𝐷 [𝑊], then there are at least five edges between {𝑣 1 , 𝑣 2 } and the set of blue vertices which are not UC-special. With the structure of the residual graph 𝐺 𝐷 given by Subclaims 6.37.1.1, 6.37.1.2, and 6.37.1.3, we apply the following discharging arguments that distribute the weights of the vertices. Discharging Rules. We initially assign charges to the (non-red) vertices of 𝐺 𝐷 so that every white vertex receives a weight of 508, and every (blue) vertex in 𝐵3 , 𝐵2 , and 𝐵1 receives a weight of 268, 236, and 194, respectively. Since Δ𝑊 (𝐵) ≤ 3, the

Section 6.2. Bounds on the Domination Number

165

sum of the charges equals w(𝐺 𝐷 ). Thereafter, we distribute the charge of every blue vertex that is not a UC-special vertex equally amongst its white neighbors, while we distribute the charge of a UC-special vertex unequally amongst its two neighbors by giving a charge of 194 to its (unique) neighbor in 𝑊0 and a charge of 42 to its other neighbor. Thus, • Every vertex in 𝐵3 gives 89 31 to each of its three (white) neighbors. • Every non-special or C-special vertex in 𝐵2 gives 118 to both its (white) neighbors. • Every UC-special vertex in 𝐵2 gives 194 to its (white) neighbor in 𝑊0 and gives 42 to its other (white) neighbor. • Every vertex in 𝐵1 gives 194 to its (unique white) neighbor. Subclaim 6.37.1.4 The charge of each component in 𝐺 𝐷 [𝑊] and that of a path 𝑃 ∈ P ★ are given by the values in Table 6.4.

component

𝐶5

𝐶4

uncovered 𝑃2

uncovered 𝑃1

charge

≥ 3380

≥ 3461 13

≥ 1672 23

≥ 1672

component

covered 𝑃2

covered 𝑃1

path 𝑃 ∈ P ★

charge

≥ 1588

≥ 1216

≥ 4020

Table 6.4 The charge of each component in 𝐺 𝐷 [𝑊] Let 𝑝 = |P ★ | and let 𝑐 4 , 𝑐 5 , 𝑝 1 , and 𝑝 2 be the number of 𝐶4 -, 𝐶5 -, uncovered 𝑃1 -, and uncovered 𝑃2 -components of 𝐺 𝐷 [𝑊], respectively. Let 𝐴′ be a 𝛾-set of the uncovered components of 𝐺 [𝑊] and let 𝐴′′ be the set of the covered blue vertices in 𝐺 𝐷 . We note that 𝐴 = 𝐴′ ∪ 𝐴′′ dominates all white vertices of 𝐺 𝐷 and | 𝐴| = 2𝑝 + 2𝑐 5 + 2𝑐 4 + 𝑝 1 + 𝑝 2 . The set 𝐷 ′ = 𝐴 ∪ 𝐷 is a dominating set of 𝐺, and so w(𝐺 𝐷 ′ ) = 0. Thus, 𝜓( 𝐴) = w(𝐺 𝐷 ) − w(𝐺 𝐷 ′ ) = w(𝐺 𝐷 ). By the discharging rules defined earlier, the weight of the blue vertices in 𝐺 𝐷 is distributed among the white vertices. Thus, by Subclaim 6.37.1.4, 𝜓( 𝐴) = w(𝐺 𝐷 ) ≥ 4020𝑝 + 3380𝑐 5 + 3461 13 𝑐 4 + 1672 23 𝑝 2 + 1672𝑝 1 ≥ 1672(2𝑝 + 2𝑐 4 + 2𝑐 5 + 𝑝 1 + 𝑝 2 ) = 1672| 𝐴|. The set 𝐴 is therefore a 𝐷-desirable set, contradicting our supposition that no such set exists.

166

Chapter 6. Upper Bounds in Terms of Minimum Degree

We now return to the proof of Theorem 6.37. Let 𝐷 0 = ∅ and let 𝐺 0 = 𝐺 𝐷0 . Note that 𝑉 (𝐺 0 ) = 𝑊 and w(𝐺 0 ) = 508𝑛. Further, 𝐵 = 𝑅 = ∅. In particular, we note that 𝐵6 = ∅. By Claim 6.37.1, there exists a 𝐷 0 -desirable set 𝐴1 . If we let 𝐷 1 = 𝐷 0 ∪ 𝐴1 = 𝐴1 and 𝐺 1 = 𝐺 𝐷1 , then the set 𝐵6 remains empty in 𝐺 1 and w(𝐺 0 ) − w(𝐺 1 ) ≥ 1672| 𝐴1 |, that is, w(𝐺 1 ) ≤ w(𝐺 0 ) − 1672| 𝐴1 |. If w(𝐺 1 ) > 0, then by Claim 6.37.1, there exists a 𝐷 1 -desirable set 𝐴2 . If we let 𝐷 2 = 𝐴1 ∪ 𝐴2 and 𝐺 2 = 𝐺 𝐷2 , then the set 𝐵6 remains empty in 𝐺 2 and w(𝐺 1 ) − w(𝐺 2 ) ≥ 1672| 𝐴2 |, that is, w(𝐺 2 ) ≤ w(𝐺 1 ) − 1672| 𝐴2 |. If w(𝐺 2 ) > 0, then we continue this process. In this way, we obtain a sequence of residual graphs 𝐺 0 , 𝐺 1 , . . . , 𝐺 𝑘 and a dominating set 𝐷 = 𝐴1 ∪ · · · ∪ 𝐴 𝑘 of 𝐺 such that 𝑘 ∑︁ | 𝐴𝑖 | 0 = w(𝐺 𝑘 ) ≤ w(𝐺 𝑘−1 ) − 1672| 𝐴 𝑘 | ≤ w(𝐺 0 ) − 1672 𝑖=1

= 508𝑛 − 1672|𝐷|. Consequently, 𝛾(𝐺) ≤ |𝐷| ≤

508 127 𝑛= 𝑛. 1672 418

The upper bound on the domination number of a graph with minimum degree at least 6 given in Theorem 6.37 is currently the best known bound. However, it is not known if this bound is achievable. Henning [460] posed the following conjecture. Conjecture 6.38 ([460]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then 𝛾(𝐺) ≤ 14 𝑛. If 𝐺 is the graph obtained from a complete graph 𝐾8 by removing the edges of a perfect matching, then 𝐺 is a 6-regular graph of order 𝑛 = 8 satisfying 𝛾(𝐺) = 2 = 14 𝑛. Hence, if Conjecture 6.38 is correct, then the bound is best possible. As a further example given in [460], if 𝐺 is the 6-regular graph of order 𝑛 = 16 shown in Figure 6.15, then 𝛾(𝐺) = 4 = 14 𝑛, where the four highlighted vertices form a 𝛾-set of 𝐺. Unfortunately, there is a large gap between the best known lower and upper bounds to date on the domination number of a graph with minimum degree at least 6. A
natural 3 1 question is whether the best known upper bound of 𝛾(𝐺) ≤ 127 418 𝑛 < 10 + 261 𝑛 given in Theorem 6.37 can be improved ever-so-slightly to the more aesthetically pleasing 3 bound 𝛾(𝐺) ≤ 10 𝑛. If so, can the vertex weighting arguments and discharging methods presented in the proof of Theorem 6.37 be sufficiently refined to achieve such an improved bound? Or is a radically new idea needed to achieve a further breakthrough yielding an improved bound?

Section 6.2. Bounds on the Domination Number

167

Figure 6.15 A 6-regular graph of order 𝑛 = 16 with 𝛾(𝐺) = 4 = 14 𝑛

We summarize the known upper bounds to date on the domination number of a graph 𝐺 with small minimum degree 𝛿 ∈ [6] in terms of its order 𝑛 in Table 6.5. As remarked earlier, the bounds for minimum degree 𝛿 ∈ {1, 2, 3} are tight, while the bounds for minimum degree 𝛿 ∈ {4, 5, 6} are currently the best known bounds to date. However, it is not known if the given bounds for minimum degree 𝛿 ∈ {4, 5, 6} are achievable. Year

𝛿(𝐺) ≥

1962

𝛿(𝐺) ≥ 1

𝛾(𝐺) ≤

1973

𝛿(𝐺) ≥ 2

𝛾(𝐺) ≤

1996

𝛿(𝐺) ≥ 3

𝛾(𝐺) ≤

2009

𝛿(𝐺) ≥ 4

𝛾(𝐺) ≤

2021

𝛿(𝐺) ≥ 5

𝛾(𝐺) ≤

2021 a b

𝛿(𝐺) ≥ 6

𝛾(𝐺) ≤

𝛾(𝐺) ≤

1 a 2𝑛 2 b 5𝑛 3 8𝑛 4 11 𝑛 1 3𝑛 127 418 𝑛

Citation [622] [79, 586] [655] [682] [123]


10778 = 𝑠, and so condition (b) holds. Further, 𝑏 6 ≤ 𝑎, 𝑏 6 − 𝑏 5 = 99, 𝑏 5 − 𝑏 4 = 127, 𝑏 4 − 𝑏 3 = 170, 𝑏 3 − 𝑏 2 = 252, and 𝑏 2 − 𝑏 1 = 397. Hence, condition (a) holds. Thus, by Theorem 6.39, 𝛾(𝐺) ≤

𝑎 3404 1702 𝑛= 𝑛= 𝑛. 𝑠 10778 5389

This is precisely the result stated formally in Theorem 6.36. More generally, Bujtás and Klavžar [125] computed the upper bounds on 𝛾(𝐺) for graphs 𝐺 with minimum degree 5 ≤ 𝛿 ≤ 50. In Table 6.7, we present their computed bounds for values of 𝛿 ∈ {7, 8, . . . , 15}, where we list in this table the values 𝑎𝑠 obtained by applying Theorem 6.39. For example, if 𝛿(𝐺) ≥ 7, then by Table 6.7, we have 𝛾(𝐺) ≤ 0.2926 𝑛, while if 𝛿(𝐺) ≥ 8, then 𝛾(𝐺) ≤ 0.2732 𝑛. The upper bounds on the domination number of

Section 6.3. Bounds on the Total Domination Number 𝛿(𝐺)

7

8

9

10

11

12

169 13

14

15

𝛾(𝐺) ≤ 0.2926 0.2732 0.2565 0.2421 0.2294 0.2182 0.2082 0.1992 0.1910 Table 6.7 Upper bounds on 𝛾(𝐺) in terms of its order with given minimum degree 𝛿(𝐺)

a graph given in Table 6.7 are currently the best known bounds on the domination number of a graph with minimum degree 𝛿 where 𝛿 ∈ {7, 8, . . . , 15}. However, it is not known if these bounds are achievable.

6.3

Bounds on the Total Domination Number

In this section, we present the bound 𝛾t (𝐺) ≤ 47 𝑛 of Henning on the total domination number of a connected graph 𝐺 of order 𝑛 ≥ 11 and minimum degree 𝛿(𝐺) ≥ 2 using the proof technique of 47 -minimal graphs. We present the pleasing result due to Archdeacon et al. that if 𝐺 is a graph of order 𝑛 with minimum degree 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 12 𝑛 using counting arguments and Brooks’ Coloring Theorem. We discuss the interplay between total domination in graphs and transversals in hypergraphs, an idea first explored by Thomassé and Yeo. Using this powerful proof technique, we present their result that if 𝐺 is a graph of order 𝑛 with minimum degree 𝛿(𝐺) ≥ 4, then 𝛾t (𝐺) ≤ 37 𝑛. We close this section with a heuristic algorithm due to Henning and Yeo that yields an upper bound on the total domination of a graph in terms of its order and minimum degree, namely if 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) = 𝛿 ≥ 1, 𝛿)
then 𝛾t (𝐺) ≤ 1+ln( 𝑛. 𝛿

6.3.1

Minimum Degree One

The situation for minimum degree one is handled in Chapter 4. We repeat the results here for completeness. Theorem 6.40 ([182]) If 𝐺 is a connected graph of order 𝑛 ≥ 3, then 𝛾t (𝐺) ≤ 23 𝑛. Theorem 6.41 ([117]) If 𝐺 is a connected graph of order 𝑛 ≥ 3, then 𝛾t (𝐺) = 23 𝑛 if and only if 𝐺 is 𝐶3 , 𝐶6 , or 𝐹 ◦ 𝑃2 for some connected graph 𝐹.

6.3.2

Minimum Degree Two

If 𝐺 is a graph of order 𝑛, each component of which is a 3-cycle or a 6-cycle, then 𝛾t (𝐺) = 23 𝑛. Hence, the upper bound in Theorem 6.40 is best possible for graphs of minimum degree 2. However, the upper bound can be improved if we impose the additional restriction that the graph 𝐺 is connected, as shown in 1995 by Sun [697]. Theorem 6.42 ([697]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then

170

Chapter 6. Upper Bounds in Terms of Minimum Degree 𝛾t (𝐺) ≤

4

7 (𝑛

 + 1) .

In 2000 Henning [453] showed that the upper bound in Theorem 6.42 can be improved slightly if we forbid six graphs of small orders. In order to state this result, we shall need the following definition. Definition 6.43 ([453]) A graph of order 𝑛 ≥ 3 is a 47 -minimal graph if 𝐺 is edge minimal with respect to satisfying the following three conditions: (a) 𝛿(𝐺) ≥ 2. (b) 𝐺 is connected. (c) 𝛾t (𝐺) ≥ 47 𝑛. Let Gtdom be the family of graphs 𝐺 that can be constructed from a connected graph 𝐹 of order at least 2 as follows: for each vertex 𝑣 of 𝐹, add a 6-cycle 𝐶𝑣 and join 𝑣 either to one vertex of 𝐶𝑣 or to two vertices at distance 2 on the cycle 𝐶𝑣 . We call the graph 𝐹 the underlying graph of 𝐺. An example of a graph 𝐺 in the family Gtdom , in the case when the underlying graph 𝐹 of 𝐺 is a 4-cycle, is illustrated in Figure 6.16. 𝐹 = 𝐶4

Figure 6.16 A graph in the family Gtdom Let 𝐺 ∈ Gtdom and let 𝐹 be the underlying graph used to construct 𝐺. For each vertex 𝑣 of 𝐹, the subgraph of 𝐺 induced by 𝑣 and the vertices of its associated 6-cycle is called a unit of 𝐺. There are two types of units. The first type of unit is obtained from a 6-cycle by adding a new vertex and joining it to two vertices at distance 2 on the cycle. In this case, we call the resulting graph 𝐺 7 , which is shown in Figure 6.17(a), and the resulting unit a 𝐺 7 -unit. The second type of unit we call a key unit, which is obtained from a 6-cycle by adding a new vertex and joining it to exactly one vertex on the cycle, as shown in Figure 6.17(b). For example, the graph in the family Gtdom shown in Figure 6.16 contains four units, two of which are key units and two of which are 𝐺 7 -units. We note that if 𝐺 ∈ Gtdom has order 𝑛 and 𝑘 units, then 𝑛 = 7𝑘. Every TD-set of the graph 𝐺 ∈ Gtdom must contain at least four vertices from each unit of 𝐺, implying that 𝛾t (𝐺) ≥ 4𝑘. However, every four consecutive vertices from the cycle 𝐶𝑣 , starting at a neighbor of 𝑣 on 𝐶𝑣 , totally dominates the vertices of the unit containing 𝑣, implying that 𝛾t (𝐺) ≤ 4𝑘. Consequently, 𝛾t (𝐺) = 4𝑘. This yields the following observation. Proposition 6.44 If 𝐺 ∈ Gtdom has order 𝑛, then 𝛾t (𝐺) = 47 𝑛.

Section 6.3. Bounds on the Total Domination Number

(a) A 𝐺 7 -unit

171

(b) A key unit

Figure 6.17 The two types of units in a graph in Gtdom

If the underlying graph 𝐹 of a graph 𝐺 ∈ Gtdom is a tree and if every unit of 𝐺 is a key unit, then we denote the resulting subfamily by Gtdom,tree . A graph 𝐺 ∈ Gtdom,tree whose underlying tree 𝐹 is a path 𝑃3 is shown in Figure 6.18.

Figure 6.18 A graph in the family Gtdom,tree Let Etdom = {𝐶3 , 𝐶5 , 𝐶6 , 𝐶7 , 𝐶10 , 𝐶14 } ∪ {𝐺 7 } be the family of seven exceptional graphs (consisting of six small cycles and an additional graph of order seven). We are now in a position to present the characterization of 47 -minimal graphs given in [453]. Since the proof of Theorem 6.45 given in [453] uses the same approach employed by McCuaig and Shepherd [586] to prove Theorem 6.16, we omit the proof. Theorem 6.45 ([453]) A graph 𝐺 is a 47 -minimal graph if and only if 𝐺 ∈ Etdom ∪ Gtdom,tree . ′ and 𝐶 ′′ be the two graphs that are obtained from a 10-cycle by adding Let 𝐶10 10 one or two chords as shown in Figure 6.19(a) and (b), respectively.

′ (a) 𝐶10

′′ (b) 𝐶10

′ and 𝐶 ′′ Figure 6.19 The graphs 𝐶10 10

We next present an improvement of the upper bound in Theorem 6.42. Let  ′ ′′ Btdom = 𝐶3 , 𝐶5 , 𝐶6 , 𝐶10 , 𝐶10 , 𝐶10

172

Chapter 6. Upper Bounds in Terms of Minimum Degree

be the family of four small cycles and the two exceptional graphs obtained from a 10-cycle by adding one or two chords. Theorem 6.46 ([453]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then 𝛾t (𝐺) ≤ 74 𝑛, unless 𝐺 is one of the six exceptional graphs in the family Btdom . Proof Let 𝐺 be a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2. Let 𝐺 ′ be obtained from 𝐺 by deleting edges, if necessary, so that the resulting graph is edge minimal with respect to the two conditions: (i) 𝐺 ′ is connected and (ii) 𝛿(𝐺 ′ ) ≥ 2. Since removing edges from a graph cannot decrease its total domination number, 𝛾t (𝐺) ≤ 𝛾t (𝐺 ′ ). If 𝐺 ′ is not a 47 -minimal graph, then 𝛾t (𝐺 ′ ) < 47 𝑛, implying that 𝛾(𝐺) ≤ 47 𝑛. If 𝐺 ′ is a 47 -minimal graph, then by Theorem 6.45, we have 𝐺 ′ ∈ Etdom ∪ Gtdom . If 𝐺 ′ ∈ Gtdom , then by Proposition 6.44, we have 𝛾(𝐺 ′ ) = 47 𝑛, while if 𝐺 ′ ∈ {𝐶7 , 𝐶14 , 𝐺 7 }, then 𝛾(𝐺 ′ ) = 47 𝑛, and in these cases we once again have 𝛾(𝐺) ≤ 47 𝑛. If 𝐺 ′ = 𝐶3 , then 𝐺 = 𝐶3 and 𝛾t (𝐺) = 2 = 17 (4𝑛 + 2). If 𝐺 ′ = 𝐶5 , then either 𝐺 = 𝐶5 and 𝛾t (𝐺) = 3 = 17 (4𝑛 + 1) or 𝛾t (𝐺) = 2 < 47 𝑛. If 𝐺 ′ = 𝐶6 , then either 𝐺 = 𝐶6  and 𝛾t (𝐺) = 4 = 47 (𝑛 + 1) or 𝛾t (𝐺) ≤ 3 < 47 𝑛. If 𝐺 ′ = 𝐶10 , then either 𝐺 ∈ 𝐶10 , ′ , 𝐶 ′′ and 𝛾 (𝐺) = 6 = 1 (4𝑛 + 2) or 𝛾 (𝐺) ≤ 5 < 4 𝑛. Thus, 𝛾 (𝐺) ≤ 4 𝑛, unless 𝐶10 t t t 7 7 7  10 ′ , 𝐶 ′′ = B 𝐺 ∈ 𝐶3 , 𝐶5 , 𝐶6 , 𝐶10 , 𝐶10 . tdom 10 Using the characterization of 47 -minimal graphs given in Theorem 6.45, the graphs of large order achieving equality in the upper bound of Theorem 6.46 were characterized by Henning in [453]. Theorem 6.47 ([453]) If 𝐺 is a connected graph of order 𝑛 ≥ 11 with 𝛿(𝐺) ≥ 2, then 𝛾t (𝐺) ≤ 47 𝑛, with equality if and only if 𝐺 = 𝐶14 or 𝐺 ∈ Gtdom .

6.3.3

Minimum Degree Three

In 2000 Favaron et al. [282] showed that the 47 -bound on the total domination number 7 given in Theorem 6.46 can be improved to a 13 -bound if the minimum degree requirement is increased to at least 3. Theorem 6.48 ([282]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 7 𝛾t (𝐺) ≤ 13 𝑛. However, it was conjectured in [282] that the total domination number of a graph with minimum degree at least 3 is at most one-half its order. The authors in [282] wrote that if the conjecture is true, then the bound is tight as may be seen as follows. Consider the generalized Petersen graph 𝑃(8, 3) shown in Figure 6.20(a). Proposition 6.49 If 𝐺 = 𝑃(8, 3), then 𝐺 has order 𝑛 = 16 and 𝛾t (𝐺) = 12 𝑛. Proof Let 𝐺 = 𝑃(8, 3), and let 𝑋 and 𝑌 be the partite sets of the bipartite graph 𝐺, as illustrated in Figure 6.20(b), where the vertices in 𝑋 and 𝑌 are colored red and blue, respectively. Thus, |𝑋 | = |𝑌 | = 8. Let 𝐷 be an arbitrary TD-set of 𝐺, and let 𝐷 𝑋 = 𝐷 ∩ 𝑋 and let 𝐷𝑌 = 𝐷 ∩ 𝑌 . The vertices in 𝑋 and 𝑌 are totally dominated by the sets 𝐷𝑌 and 𝐷 𝑋 , respectively.

Section 6.3. Bounds on the Total Domination Number

173

𝑦2 𝑥1

𝑥2 𝑦1

(a)

(b)

Figure 6.20 The generalized Petersen graph 𝑃(8, 3)

Suppose that |𝐷 𝑋 | ≤ 3. Since |𝑌 | = 8 and each vertex totally dominates three vertices, we have |𝐷 𝑋 | = 3. If every two vertices in 𝐷 𝑋 have a common neighbor, then 𝐷 𝑋 would totally dominate at most seven vertices in 𝑌 , a contradiction. Hence, the set 𝐷 𝑋 contains two vertices 𝑥1 and 𝑥2 that have no common neighbor in 𝑌 . Every vertex in 𝑋 has a common neighbor with every other vertex in 𝑋, except for the unique vertex in 𝑋 at distance 4 from it. Thus, 𝑑𝐺 (𝑥 1 , 𝑥2 ) = 4. By symmetry, we may assume that 𝑥1 and 𝑥2 are the vertices labeled in Figure 6.20(b). The two vertices in 𝑌 not totally dominated by {𝑥1 , 𝑥2 } are at distance 4 in 𝐺. These vertices are labeled 𝑦 1 and 𝑦 2 in Figure 6.20(b). The third vertex of 𝐷 𝑋 , which is different from 𝑥1 and 𝑥2 , therefore cannot totally dominate both 𝑦 1 and 𝑦 2 , contradicting the fact that 𝐷 𝑋 totally dominates the set 𝑌 . Hence, |𝐷 𝑋 | ≥ 4. Using similar arguments, |𝐷𝑌 | ≥ 4. Hence, |𝐷| = |𝐷 𝑋 | + |𝐷𝑌 | ≥ 4 + 4 = 8. Since 𝐷 is an arbitrary TD-set of 𝐺, this implies that 𝛾t (𝐺) ≥ 8. The set of vertices of the outer 8-cycle is a TD-set of 𝐺, implying that 𝛾t (𝐺) ≤ 8. Consequently, 𝛾t (𝐺) = 8 = 12 𝑛. The authors in [282] constructed the following two infinite families Gcubic and Hcubic of cubic graphs with total domination numbers one-half their orders. For 𝑘 ≥ 1, let 𝐺 𝑘 be the graph constructed as follows. Consider two copies of the path 𝑃2𝑘 with respective vertex sequences 𝑎 1 𝑏 1 𝑎 2 𝑏 2 . . . 𝑎 𝑘 𝑏 𝑘 and 𝑐 1 𝑑1 𝑐 2 𝑑2 . . . 𝑐 𝑘 𝑑 𝑘 . For each 𝑖 ∈ [𝑘], join 𝑎 𝑖 to 𝑑𝑖 and 𝑏 𝑖 to 𝑐 𝑖 . To complete the construction of the graph 𝐺 𝑘 join 𝑎 1 to 𝑐 1 and 𝑏 𝑘 to 𝑑 𝑘 . We note that 𝐺 1 = 𝐾4 . Let Gcubic = {𝐺 𝑘 : 𝑘 ≥ 1}. For 𝑘 ≥ 2, let 𝐻 𝑘 be obtained from 𝐺 𝑘 by deleting the two edges 𝑎 1 𝑐 1 and 𝑏 𝑘 𝑑 𝑘 and adding the two edges 𝑎 1 𝑏 𝑘 and 𝑐 1 𝑑 𝑘 . Let Hcubic = {𝐻 𝑘 : 𝑘 ≥ 2}. The graphs 𝐺 4 ∈ Gcubic and 𝐻4 ∈ Hcubic are illustrated in Figure 6.21(a) and (b), respectively. Proposition 6.50 If 𝐺 ∈ Gcubic ∪ Hcubic has order 𝑛, then 𝛾t (𝐺) = 12 𝑛. Proof Let 𝐺 ∈ Gcubic have order 𝑛, and so 𝐺 = 𝐺 𝑘 for some 𝑘 ≥ 1. Thus, 𝑛 = 4𝑘. Let 𝑉 (𝐺) = 𝑉1 ∪𝑉2 ∪ · · · ∪𝑉𝑘 , where 𝑉𝑖 = {𝑎 𝑖 , 𝑏 𝑖 , 𝑐 𝑖 , 𝑑𝑖 } for 𝑖 ∈ [𝑘]. The set {𝑎 1 , 𝑎 2 , . . . , 𝑎 𝑘 } ∪ {𝑏 1 , 𝑏 2 , . . . , 𝑏 𝑘 } is a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ 2𝑘. Hence, it suffices for us to show that 𝛾t (𝐺) ≥ 2𝑘. Suppose, to the contrary, that 𝛾t (𝐺) < 2𝑘. Thus, for

Chapter 6. Upper Bounds in Terms of Minimum Degree

174 𝑎1

𝑐1

𝑏1

𝑑1

𝑎2

𝑐2

𝑏2

𝑑2

𝑎3

𝑐3

𝑏3

𝑑3

𝑎4

𝑐4 𝑑4

𝑏4 (a) 𝐺 4

(b) 𝐻4

Figure 6.21 The graphs 𝐺 4 ∈ Gcubic and 𝐻4 ∈ Hcubic

every 𝛾t -set 𝑋 of 𝐺, we have |𝑋 ∩ 𝑉𝑖 | ≤ 1 for at least one 𝑖 ∈ [𝑘]. For each such set 𝑋, let  𝐼 𝑋 = 𝑖 ∈ [𝑘] : |𝑋 ∩ 𝑉𝑖 | ≤ 1 . Among all 𝛾t -sets of 𝐺, let 𝑋 be chosen so that |𝐼 𝑋 | is a minimum. By supposition, |𝐼 𝑋 | ≥ 1, that is, 𝑖 ∈ 𝐼 𝑋 for at least one 𝑖 ∈ [𝑘]. Suppose firstly that |𝑋 ∩ 𝑉1 | ≤ 1. If 𝑋 ∩ 𝑉1 = ∅, then 𝑎 1 and 𝑐 1 are not totally dominated by the set 𝑋, a contradiction. Hence, |𝑋 ∩ 𝑉1 | ≥ 1. Suppose that |𝑋 ∩ 𝑉1 | = 1. In order to totally dominate the vertices 𝑎 1 and 𝑐 1 , either 𝑏 1 or 𝑑1 belongs to the set 𝑋. By symmetry, we may assume that 𝑏 1 ∈ 𝑋. In order to totally dominate the vertices 𝑏 1 and 𝑑1 , we have 𝑎 2 ∈ 𝑋 and 𝑐 2 ∈ 𝑋, respectively. In order to totally dominate vertex 𝑐 2 , at least one of 𝑏 2 or 𝑑2
belongs to the set 𝑋. But then 𝑌 = 𝑋 \ {𝑎 2 } ∪ {𝑎 1 } is a 𝛾t -set of 𝐺 with |𝐼𝑌 | < |𝐼 𝑋 |, a contradiction. Hence, |𝑋 ∩ 𝑉1 | ≥ 2, and so 𝑖 ∉ 𝐼 𝑋 . By symmetry, 𝑘 ∉ 𝐼 𝑋 . Thus, 2 ≤ 𝑖 ≤ 𝑘 − 1. Suppose that 𝑋 ∩ 𝑉𝑖 = ∅. In order to totally dominate the vertices in the set 𝑉𝑖 , we have {𝑏 𝑖−1 , 𝑐 𝑖−1 , 𝑎 𝑖+1 , 𝑑𝑖+1 } ⊆ 𝑋. In order to totally dominate 𝑏 𝑖−1 , at least one of 𝑎 𝑖−1 or 𝑐 𝑖−1 belongs to the set 𝑋. In order to totally dominate vertex 𝑎
𝑖+1 , at least one of 𝑏 𝑖+1 or 𝑑𝑖+1 belongs to the set 𝑋. But then 𝑌 = 𝑋 \ {𝑎 𝑖+1 , 𝑏 𝑖−1 } ∪ {𝑎 𝑖 , 𝑏 𝑖 } is a 𝛾t -set of 𝐺 with |𝐼𝑌 | < |𝐼 𝑋 |, a contradiction. Hence, |𝑋 ∩ 𝑉𝑖 | ≥ 1, implying by supposition that |𝑋 ∩ 𝑉𝑖 | = 1. By symmetry we may assume that 𝑋 ∩ 𝑉𝑖 = {𝑎 𝑖 }. In order to totally dominate vertex 𝑐 𝑖 , we have 𝑑𝑖−1 ∈ 𝑋. In order to totally dominate vertex 𝑑𝑖−1 , at least one of 𝑎 𝑖−1 or 𝑐 𝑖−1 belongs to the set 𝑋. In
order to totally dominate vertex 𝑎 𝑖 , we have 𝑏 𝑖−1 ∈ 𝑋. But then 𝑌 = 𝑋 \ {𝑏 𝑖−1 } ∪ {𝑏 𝑖 } is a 𝛾t -set of 𝐺 with |𝐼𝑌 | < |𝐼 𝑋 |, a contradiction. We deduce, therefore, that 𝛾t (𝐺) ≥ 2𝑘. As observed earlier, 𝛾t (𝐺) ≤ 2𝑘. Consequently, 𝛾t (𝐺) = 2𝑘 = 12 𝑛. Let 𝐺 ∈ Hcubic have order 𝑛, and so 𝐺 = 𝐻 𝑘 for some 𝑘 ≥ 2. The proof in this case is similar to the previous case, except simpler since we need only consider the case when |𝑋 ∩ 𝑉𝑖 | ≤ 1 for some 𝑖 ∈ [𝑘] (and not consider separately the case when |𝑋 ∩ 𝑉1 | ≤ 1 or |𝑋 ∩ 𝑉𝑘 | ≤ 1, due to the symmetry of 𝐺).

Section 6.3. Bounds on the Total Domination Number

175

This conjecture in [282] attracted considerable interest. In 2004 a group of mathematicians from the USA, the Czech Republic, China, and Israel, combined forces and presented the following remarkable proof of this conjecture that the total domination of a graph with minimum degree at least 3 is at most one-half its order. In order to prove this result, Archdeacon et al. [35] surprisingly used Brooks’ Coloring Theorem. The key result they use to prove the conjecture is the following lemma. Lemma 6.51 ([35]) If 𝐻 is a bipartite graph with partite sets 𝑋 and 𝑌 whose vertices in 𝑌 are of degree at least 3, then there exists a set 𝐴 ⊆ 𝑋 such that 𝐴 dominates 𝑌 and | 𝐴| ≤ 41 |𝑋 ∪ 𝑌 |. Proof Let |𝑋 | = 𝑥 and |𝑌 | = 𝑦. We proceed by induction on |𝑉 (𝐻)| + |𝐸 (𝐻)|. The smallest graph described by the lemma is 𝐻 = 𝐾1,3 , where 𝑌 consists of the central vertex of the star and 𝑋 the set of three leaves. In this case, choosing the set 𝐴 to consist of any arbitrary vertex in 𝑋 yields the desired result. This establishes the base case. For the inductive hypothesis, assume the result is true for all bipartite graphs 𝐻 ′ with partite sets (𝑋 ′ , 𝑌 ′ ) whose vertices in 𝑌 ′ are of degree at least 3 in 𝐻 ′ . In what follows, we let |𝑋 ′ | = 𝑥 ′ and |𝑌 ′ | = 𝑦 ′ . Suppose there exists a vertex 𝑣 ∈ 𝑌 of degree at least 4. Let 𝑒 be any edge incident with 𝑣 and consider the bipartite graph 𝐻 ′ = 𝐻 − 𝑒 with partite sets (𝑋 ′ , 𝑌 ′ ), where 𝑋 ′ = 𝑋 and 𝑌 ′ = 𝑌 . We note that every vertex in 𝑌 ′ has degree at least 3 in 𝐻 ′ . Applying the inductive hypothesis to 𝐻 ′ , there exists a subset 𝐴′ ⊆ 𝑋 ′ such that 𝐴′ dominates 𝑌 ′ and | 𝐴′ | ≤ 14 (𝑥 ′ + 𝑦 ′ ) = 14 (𝑥 + 𝑦). The set 𝐴 = 𝐴′ is the desired set. Hence, we may assume that every vertex in 𝑌 has degree exactly 3 in 𝐻. Suppose there exists an isolated vertex 𝑣 ∈ 𝑋. Consider the bipartite graph 𝐻 ′ = 𝐻 − 𝑣 with partite sets (𝑋 ′ , 𝑌 ′ ), where 𝑋 ′ = 𝑋 \ {𝑣} and 𝑌 ′ = 𝑌 . Applying the inductive hypothesis to 𝐻 ′ , there exists a subset 𝐴′ ⊆ 𝑋 ′ such that 𝐴′ dominates 𝑌 ′ and | 𝐴′ | ≤ 14 (𝑥 ′ + 𝑦 ′ ) < 14 (𝑥 + 𝑦). The set 𝐴 = 𝐴′ is the desired set. Hence, we may assume that every vertex in 𝑋 has degree at least 1 in 𝐻. Suppose there exists a vertex 𝑣 in 𝑋 of degree at least 3. If 𝑣 dominates 𝑌 , then the desired result is immediate by choosing 𝐴 = {𝑣} and noting that 𝑥 ≥ 3 and 𝑦 ≥ 3, and so | 𝐴| < 14 (𝑥 + 𝑦). Hence, we may assume that 𝑣 does not dominate 𝑌 . We consider the bipartite graph 𝐻 ′ = 𝐻 − N[𝑣] with partite sets (𝑋 ′ , 𝑌 ′ ), where 𝑋 ′ = 𝑋 \ {𝑣} and 𝑌 ′ = 𝑌 \ N(𝑣). We note that 𝑥 ′ = 𝑥 − 1 and 𝑦 ′ ≤ 𝑦 − 3. Applying the inductive hypothesis to 𝐻 ′ , there exists a subset 𝐴′ ⊆ 𝑋 ′ such that 𝐴′ dominates 𝑌 ′ and | 𝐴′ | ≤ 14 (𝑥 ′ + 𝑦 ′ ) ≤ 14 (𝑥 + 𝑦) − 1. The set 𝐴 = 𝐴′ ∪ {𝑣} is the desired set. Hence, we may assume that every vertex in 𝑋 has degree at most 2 in 𝐻. With our current assumptions, every vertex in 𝑋 has degree 1 or 2 in 𝐻 and every vertex in 𝑌 has degree 3 in 𝐻. Let 𝑋𝑖 be the set of vertices in 𝑋 of degree 𝑖 and let |𝑋𝑖 | = 𝑥𝑖 for 𝑖 ∈ [2], and so 𝑥 = 𝑥1 + 𝑥2 and 𝑥1 + 2𝑥 2 = |𝐸 (𝐻)| = 3𝑦. Suppose that two vertices 𝑣 1 and 𝑣 2 in 𝑋2 have two common neighbors 𝑤 1 and 𝑤 2 in 𝑌 . We consider the bipartite graph 𝐻 ′ = 𝐻 − {𝑣 1 , 𝑣 2 , 𝑤 1 , 𝑤 2 } with partite sets (𝑋 ′ , 𝑌 ′ ), where 𝑋 ′ = 𝑋 \ {𝑣 1 , 𝑣 2 } and 𝑌 ′ = 𝑌 \ {𝑤 1 , 𝑤 2 }. We note that 𝑥 ′ = 𝑥 − 2 and 𝑦 ′ = 𝑦 − 2. Applying the inductive hypothesis to 𝐻 ′ , there exists a subset 𝐴′ ⊆ 𝑋 ′ such that 𝐴′ dominates 𝑌 ′ and | 𝐴′ | ≤ 14 (𝑥 ′ + 𝑦 ′ ) = 14 (𝑥 + 𝑦) − 1. The set 𝐴 = 𝐴′ ∪ {𝑣 1 }

Chapter 6. Upper Bounds in Terms of Minimum Degree

176

is the desired set. Hence, we may assume that every two vertices in 𝑋2 have at most one common neighbor in 𝑌 . Let 𝐹 be the graph with 𝑉 (𝐹) = 𝑋2 , where two vertices are adjacent in 𝐹 if and only if they have a common neighbor in 𝐻. Let 𝑣 ∈ 𝑉 (𝐹). Let 𝑤 1 and 𝑤 2 be the two neighbors of 𝑣 in 𝐻 that belong to 𝑌 , and let N 𝐻 (𝑤 𝑖 ) = {𝑢 𝑖 , 𝑣, 𝑣 𝑖 } for 𝑖 ∈ [2]. Thus, N𝐹 (𝑣) = {𝑢 1 , 𝑣 1 } ∪ {𝑢 2 , 𝑣 2 }. If {𝑢 1 , 𝑣 1 } ∩ {𝑢 2 , 𝑣 2 } = ∅, then 𝑣 is the only common neighbor of 𝑤 1 and 𝑤 2 in 𝐻, and deg𝐹 (𝑣) = 4. If {𝑢 1 , 𝑣 1 } ∩ {𝑢 2 , 𝑣 2 } ≠ ∅, then deg𝐹 (𝑣) ≤ 3. Thus, Δ(𝐹) ≤ 4. Further, if Δ(𝐹) = 4, then renaming vertices if necessary, we may assume that deg𝐹 (𝑣) = 4. However, in this case by our earlier observations, neither 𝑢 𝑖 nor 𝑣 𝑖 is adjacent to 𝑤 3−𝑖 in the graph 𝐻 for 𝑖 ∈ [2], implying that there is no edge between the vertices of {𝑢 1 , 𝑣 1 } and the vertices of {𝑢 2 , 𝑣 2 } in the graph 𝐹. Therefore, no component of 𝐹 is a complete graph 𝐾5 . Applying Brooks’ Coloring Theorem, the graph 𝐹 is 4-colorable, and so 𝐹 has an independent set 𝐼 such that |𝐼 | ≥ 41 |𝑉 (𝐹)| = 14 𝑥 2 . In the graph 𝐻, the vertices in the set 𝐼 have vertex-disjoint neighborhoods, and so |N 𝐻 (𝐼)| = 2|𝐼 |. We now consider the set 𝑌 = 𝑌 \ N 𝐻 (𝐼). For each vertex 𝑤 ∈ 𝑌 , we choose an adjacent vertex 𝑤 ′ ∈ 𝑋 and we let Ø 𝐼′ = {𝑤 ′ }, 𝑤 ∈𝑌

𝐼′

|𝐼 ′ |

and so ⊆ 𝑋 \ 𝐼 and ≤ |𝑌 | = 𝑦 − 2|𝐼 |. We now let 𝐴 = 𝐼 ∪ 𝐼 ′ . Since 𝑥 = 𝑥1 + 𝑥2 5 and 𝑥1 + 2𝑥2 = |𝐸 (𝐻)| = 3𝑦, we have 𝑦 = 13 (𝑥1 + 2𝑥 2 ) and 14 (𝑥 + 𝑦) = 13 𝑥1 + 12 𝑥2 . Thus, | 𝐴| = |𝐼 | + |𝐼 ′ | ≤ 𝑦 − |𝐼 | ≤ 𝑦 − 14 𝑥2 = 13 (𝑥 1 + 2𝑥2 ) − 14 𝑥2 = 13 𝑥1 +

5 12 𝑥 2

= 14 (𝑥 + 𝑦).

By construction, the set 𝐴 ⊆ 𝑋 dominates the set 𝑌 in the graph 𝐻. As a consequence of Lemma 6.51, we have the main result of Archdeacon et al. [35]. Theorem 6.52 ([35]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 12 𝑛. Proof Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, and let 𝑉 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 }. Let 𝐻 be the bipartite graph with partite sets 𝑋 = {𝑥1 , 𝑥2 , . . . , 𝑥 𝑛 } and 𝑌 = {𝑦 1 , 𝑦 2 , . . . , 𝑦 𝑛 }, and where 𝑥 𝑖 𝑦 𝑗 is an edge in 𝐻 if and only if 𝑣 𝑖 𝑣 𝑗 is an edge in 𝐺, where 𝑖, 𝑗 ∈ [𝑛] and 𝑖 ≠ 𝑗. By construction, 𝛿(𝐻) = 𝛿(𝐺) ≥ 3. In particular, every vertex in 𝑌 has degree at least 3 in 𝐻. By Lemma 6.51, there exists a set 𝐴 ⊆ 𝑋 such that 𝐴 dominates 𝑌 and | 𝐴| ≤ 14 |𝑋 ∪ 𝑌 | = 12 𝑛. Every vertex 𝑦 𝑗 ∈ 𝑌 is adjacent to a vertex 𝑥𝑖 ∈ 𝐴 in 𝐻, where 𝑖, 𝑗 ∈ [𝑛] and 𝑖 ≠ 𝑗. Thus, every vertex 𝑣 𝑗 ∈ 𝑉 is adjacent to a vertex 𝑣 𝑖 ∈ 𝐴 in 𝐺, where 𝑖, 𝑗 ∈ [𝑛] and 𝑖 ≠ 𝑗. This implies that 𝐴 is a TD-set in 𝐺, and so 𝛾t (𝐺) ≤ | 𝐴| ≤ 12 𝑛. A natural problem is to characterize the connected graphs that achieve equality in the upper bound of Theorem 6.52. By Proposition 6.49, the generalized Petersen graph 𝑃(8, 3) achieves equality in the bound of Theorem 6.52, and by Proposition 6.50, we have two infinite classes of cubic graphs that achieve equality in the

Section 6.3. Bounds on the Total Domination Number

177

bound. Hence, this bound is tight. However, for several years it remained an open problem to give a complete characterization of the graphs that achieve equality in the bound of Theorem 6.52. Using a graph theoretic approach, this appears to be a difficult problem. In the next section, however, we discuss an interplay with total dominating sets in graphs and transversals in hypergraphs that will enable us to provide such a characterization, showing that the generalized Petersen graph 𝑃(8, 3) and the two infinite families, Gcubic and Hcubic , constructed in [282], are precisely the extremal graphs. The result of Theorem 6.52 was strengthened in 2007 by Lam and Wei [553]. Theorem 6.53 ([553]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 such that deg(𝑢) + deg(𝑣) ≥ 5 for every two adjacent vertices 𝑢 and 𝑣 of 𝐺, then 𝛾t (𝐺) ≤ 21 𝑛. As a special case of Theorem 6.53, we have the following result. Theorem 6.54 ([553]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 such that the set of vertices of degree 2 in 𝐺 form an independent set, then 𝛾t (𝐺) ≤ 12 𝑛. The result of Theorem 6.53 was further strengthened in 2007 by Henning and Yeo [481] who relaxed the condition that every two adjacent vertices have degree sum at least 5. In order to state their result, we introduce some additional terminology. Let 𝐺 be a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and Δ(𝐺) ≥ 3. Let L be the set of vertices of 𝐺 of degree at least 3 in 𝐺, and let 𝑃 be any path component of 𝐺 − L. If |𝑉 (𝑃)| ≡ 0 (mod 4) and either the two ends of 𝑃 are adjacent in 𝐺 to a common vertex of L or the two ends of 𝑃 are adjacent in 𝐺 to different, but adjacent, vertices of L, then we call 𝑃 a 0★-path. If |𝑉 (𝑃)| ≥ 5 and |𝑉 (𝑃)| ≡ 1 (mod 4) with the two ends of 𝑃 adjacent in 𝐺 to a common vertex of L, we call 𝑃 a 1★-path. If |𝑉 (𝑃)| ≡ 3 (mod 4), we call 𝑃 a 3★-path. For 𝑖 ∈ {0, 1, 3}, we denote the number of 𝑖★-paths in 𝐺 by 𝑝 𝑖 . Theorem 6.55 ([481]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and Δ(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 12 (𝑛 + 𝑝 0 + 𝑝 1 + 𝑝 3 ).

6.3.4

An Interplay with Transversals in Hypergraphs

If the minimum degree of a graph is at least 4, then establishing good upper bounds on the total domination number of a graph using a purely graph theory approach is often challenging. Thankfully, we do have a powerful tool at our disposal, namely a translation of the problem of determining the total domination number of a graph to the problem of determining the transversal number of a hypergraph that can be associated with the graph. Indeed, much of the interest in total domination in graphs dating back to 2003 arises from its interplay with transversals in hypergraphs. Before we describe this interplay between graphs and hypergraphs, which is explained more comprehensively in the book [490], we present the basic hypergraph terminology and notation that we shall use. Hypergraphs are systems of sets which are conceived as natural extensions of graphs. A hypergraph 𝐻 is a finite set 𝑉 (𝐻) of elements, called vertices, together with a finite multiset 𝐸 (𝐻) of subsets of 𝑉 (𝐻),

178

Chapter 6. Upper Bounds in Terms of Minimum Degree

called hyperedges. Abusing terminology, we often refer to a “hyperedge” of a hypergraph simply as an “edge” of the hypergraph. The notation we use for hypergraphs in what follows is similar to that we use for graphs. Specifically, the order of 𝐻 is 𝑛(𝐻) = |𝑉 (𝐻)| and the size of 𝐻 is 𝑚(𝐻) = |𝐸 (𝐻)|. A 𝑘-edge in 𝐻 is an edge of size 𝑘. The hypergraph 𝐻 is 𝑘-uniform if every edge of 𝐻 is a 𝑘-edge. Every (simple) graph is a 2-uniform hypergraph. Thus, graphs are special hypergraphs. Two vertices in 𝐻 are adjacent if they belong to a common edge of 𝐻. The degree of a vertex 𝑣 in 𝐻, denoted by deg 𝐻 (𝑣), is the number of edges of 𝐻 which contain 𝑣. The minimum and maximum degrees among the vertices of 𝐻 are denoted by 𝛿(𝐻) and Δ(𝐻), respectively. We refer the reader to Appendix A for additional hypergraph terminology. The Fano plane, shown in Figure 6.22, is an example of a 3-uniform, 3-regular hypergraph of order 𝑛(𝐻) = 7 and size 𝑚(𝐻) = 7. As with graphs, we write 𝑛 for 𝑛(𝐻) and 𝑚 for 𝑚(𝐻) when the graph 𝐻 is clear from the context.

Figure 6.22 The Fano plane A transversal (also called a hitting set in the literature) in a hypergraph 𝐻 is a set 𝑇 of vertices that have a nonempty intersection with every edge of 𝐻. Such a set 𝑇 is said to cover or hit every edge of 𝐻. The transversal number 𝜏(𝐻) of 𝐻 is the minimum cardinality of a transversal in 𝐻. We note the traversal number is called the vertex cover number when 𝐻 is a graph and is denoted by 𝛽(𝐻). A transversal in 𝐻 of cardinality 𝜏(𝐻) is called a 𝜏-transversal of 𝐻. For example, if 𝐻 is the Fano plane shown in Figure 6.22, then 𝜏(𝐻) = 3. Moreover, if 𝑒 is an arbitrary edge of 𝐻, then the set consisting of the three vertices that belong to the edge 𝑒 is a 𝜏-transversal of 𝐻. Let 𝐺 be a graph with minimum degree 𝛿 ≥ 1 and order 𝑛. Let ONH(𝐺) be the open neighborhood hypergraph of 𝐺, where the hypergraph ONH(𝐺) has the same vertex set as 𝐺, namely 𝑉, and whose edge set consists of the open neighborhoods of the vertices in the graph 𝐺. Thus, ONH(𝐺) has exactly 𝑛 edges corresponding to the open neighborhoods N𝐺 (𝑣) of the vertices 𝑣 in the graph 𝐺. By our supposition that 𝐺 has minimum degree 𝛿 ≥ 1, every open neighborhood of a vertex in 𝐺 has cardinality at least 𝛿, and therefore each edge of ONH(𝐺) has size at least 𝛿. To illustrate the open neighborhood hypergraph of a graph, if 𝐺 = 𝐺 14 is the well-known Heawood graph shown in Figure 6.23(a), then the open neighborhood hypergraph of 𝐺 is shown in Figure 6.23(b). We note that in this case, ONH(𝐺) consists of two vertex-disjoint copies of the Fano plane.

Section 6.3. Bounds on the Total Domination Number 7

𝑎

1

𝑔

179

1

𝑎

𝑏

6

2

3

6

2

𝑔

𝑏 𝑑

𝑐

𝑓 3

5 𝑒

4 (a) 𝐺 14

𝑑

7

5

4

𝑒

𝑐

𝑓

(b) ONH(𝐺 14 )

Figure 6.23 The Heawood graph and its open neighborhood hypergraph

We are now in a position to describe the interplay between total domination in graphs and transversals in hypergraphs. As before, let 𝐺 be a graph with minimum degree 𝛿 ≥ 1 and order 𝑛. Every TD-set in 𝐺 contains a vertex from the open neighborhood of every vertex in 𝐺, and is therefore a transversal in ONH(𝐺). In particular, if 𝐷 is a 𝛾t -set of 𝐺, then 𝐷 is a transversal in ONH(𝐺), and so 𝜏(ONH(𝐺)) ≤ |𝐷| = 𝛾t (𝐺). On the other hand, every transversal in ONH(𝐺) contains a vertex from the open neighborhood of every vertex of 𝐺, and is therefore a TD-set in 𝐺. In particular, if 𝑇 is a 𝜏-transversal of ONH(𝐺), then 𝑇 is a TD-set in 𝐺, and so 𝛾t (𝐺) ≤ |𝑇 | = 𝜏(ONH(𝐺)). Consequently, 𝛾t (𝐺) = 𝜏(ONH(𝐺)). Thus, the transversal number of the open neighborhood hypergraph of a graph is precisely the total domination number of the graph. We state this formally as follows. Observation 6.56 If 𝐺 is an isolate-free graph and ONH(𝐺) is the open neighborhood hypergraph of 𝐺, then 𝛾t (𝐺) = 𝜏(ONH(𝐺)). To illustrate Observation 6.56, consider our earlier example when 𝐺 = 𝐺 14 is the Heawood graph shown in Figure 6.23(a). Using purely graph theory arguments, one can show (with some work) that 𝛾t (𝐺) = 6. One can also observe, as done earlier, that the open neighborhood hypergraph ONH(𝐺) of 𝐺 consists of two components, both isomorphic to the Fano plane (see Figure 6.23(b)). If 𝐻 is the Fano plane, then 𝜏(𝐻) = 3, implying that 𝜏(ONH(𝐺)) = 6. Thus, by Observation 6.56, we can immediately deduce that 𝛾t (𝐺) = 6. The idea of using transversals in hypergraphs to obtain bounds on the total domination number of a graph first appeared in a paper by Thomassé and Yeo [709] submitted in 2003 (but only published in 2007). Their paper served to open the floodgates for further improved bounds on the total domination number of a graph with given minimum degree. Without wishing to detract from the elegant graph theoretic proof of Theorem 6.52 given in 2004 by Archdeacon et al. [35], we note that the result itself can readily be deduced from the following 1990 hypergraph result due to Tuza [720] and Chvátal and McDiarmid [176]. However, as remarked earlier, the interplay between total domination in graphs and transversals in hypergraphs seemed to pass

180

Chapter 6. Upper Bounds in Terms of Minimum Degree

by unnoticed until announced in 2003 by Thomassé and Yeo [709]. We omit a proof of Theorem 6.57, which can be found in [490]. Theorem 6.57 ([176, 720]) If 𝐻 is a hypergraph of order 𝑛 and size 𝑚 where all edges have size at least 3, then 𝜏(𝐻) ≤ 41 (𝑛 + 𝑚). As an immediate consequence of Theorem 6.57, we have the result of Theorem 6.52, which we restate as a corollary (of Theorem 6.57). Corollary 6.58 ([176, 720]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 12 𝑛. Proof Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺) ≥ 3. The open neighborhood hypergraph ONH(𝐺) of 𝐺 has 𝑛 vertices and 𝑛 edges where all edges have size at least 𝛿(𝐺) ≥ 3. Hence, by Observation 6.56 and Theorem 6.57, we have 𝛾t (𝐺) = 𝜏(ONH(𝐺)) ≤ 14 (𝑛 + 𝑚) = 12 𝑛. In order to characterize the connected graphs achieving equality in the upper bound of Theorem 6.52, in 2008 Henning and Yeo [483] first characterized the hypergraphs that achieve equality in the upper bound of Theorem 6.57. We omit the details of their characterization, which are discussed in [490]. As a consequence of this hypergraph characterization, the authors in [483] obtained the desired graph theory characterization of the extremal graphs achieving equality in the bound of Theorem 6.52. We omit the proof, which can be found in [483] and relies heavily on the interplay between total domination in graphs and transversals in hypergraphs. Theorem 6.59 ([483]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) = 12 𝑛 if and only if 𝐺 ∈ Gcubic ∪ Hcubic or 𝐺 is the generalized Petersen graph 𝑃(8, 3).

6.3.5

Minimum Degree Four

In 1992 Chvátal and McDiarmid [176] established the following upper bound on the transversal number of uniform hypergraphs in terms of their order and size. Theorem 6.60 ([176]) For every integer 𝛿 ≥ 2, if 𝐻 is a 𝛿-uniform hypergraph of order 𝑛 and size 𝑚, then   𝑛 + 2𝛿 𝑚 𝜏(𝐻) ≤  3 𝛿  . 2

As a consequence of Theorem 6.60, we have the following upper bound on the total domination number of a graph in terms of its order 𝑛 and minimum degree 𝛿. Theorem 6.61 ([176]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) = 𝛿 ≥ 2, then  ! 1 + 2𝛿 𝛾t (𝐺) ≤  3 𝛿  𝑛. 2

Section 6.3. Bounds on the Total Domination Number

181

Proof Let 𝐺 be a graph with minimum degree 𝛿(𝐺) = 𝛿 ≥ 2 and order 𝑛. Consider the hypergraph 𝐻 obtained from the open neighborhood hypergraph ONH(𝐺) by “shrinking” all edges of ONH(𝐺), if necessary, to edges of size 𝛿, that is, if 𝑒 is an edge of ONH(𝐺) of size 𝑘 for 𝑘 > 𝛿, then we select a subset 𝑒 ′ ⊂ 𝑒 of 𝑘 − 𝛿 vertices arbitrarily, and remove the vertices in 𝑒 ′ from the edge 𝑒. The resulting edge 𝑒 \ 𝑒 ′ of size 𝛿 is the edge of 𝐻 that replaces the edge 𝑒 of ONH(𝐺). By construction, the hypergraph 𝐻 is a 𝛿-uniform hypergraph with 𝑛 vertices and 𝑛 edges. Every transversal in 𝐻 is a transversal in ONH(𝐺) and so, by Observation 6.56, 𝛾t (𝐺) = 𝜏(ONH(𝐺)) ≤ 𝜏(𝐻). The desired result now follows from Theorem 6.60. In Table 6.8, we list the bounds in Theorem 6.61 for 𝛿 ∈ [10].

𝛿 𝛿 !

1+ 2  3𝛿 

𝑛

1

2

3

4

5

6

7

8

9

10

𝑛

2 3𝑛

1 2𝑛

1 2𝑛

3 7𝑛

4 9𝑛

2 5𝑛

5 12 𝑛

5 13 𝑛

2 5𝑛

2

Table 6.8 Upper bounds on 𝛾t (𝐺) in Theorem 6.61 for 𝛿 ∈ [10] The bounds in Theorem 6.61 for small 𝛿 ∈ [3] are best possible. For 𝛿 ∈ [2], this may be seen by taking 𝐺 to be the disjoint union of copies of 𝐾 𝛿+1 , while for 𝛿 = 3 the bound given in Theorem 6.61 is precisely the tight bound of Theorem 6.52. However, for 𝛿 ≥ 4, the upper bounds in Theorem 6.61 are not best possible. In their seminal 2007 paper, Thomassé and Yeo [709] improved the upper bound given in Theorem 6.61 when 𝛿 = 4. Theorem 6.62 ([709]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 4, then 𝛾t (𝐺) ≤ 37 𝑛. In order to prove Theorem 6.62, Thomassé and Yeo [709] proved the following result on transversals in 4-uniform hypergraphs. Theorem 6.63 ([709]) If 𝐻 is a 4-uniform hypergraph of order 𝑛 and size 𝑚, then 5 4 𝜏(𝐻) ≤ 21 𝑛 + 21 𝑚. Theorem 6.63 is a special case of a stronger, more general result proven in [709]. However, the statement of Theorem 6.63 suffices for our purposes. To see this, let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺) ≥ 4. The open neighborhood hypergraph ONH(𝐺) of 𝐺 has 𝑛 vertices and 𝑛 edges where all edges have size at least 𝛿(𝐺) ≥ 4. As before, we consider the hypergraph 𝐻 obtained from ONH(𝐺) by shrinking all edges of ONH(𝐺), if necessary, to edges of size 4 to produce a 4-uniform hypergraph with 𝑛 vertices and 𝑛 edges. Every transversal in 𝐻 is a transversal in ONH(𝐺) and so, by Observation 6.56 1 and Theorem 6.63, we have 𝛾t (𝐺) = 𝜏(ONH(𝐺)) ≤ 𝜏(𝐻) ≤ 21 (5𝑛 + 4𝑚) = 37 𝑛. Hence, Theorem 6.62 is an immediate consequence of Theorem 6.63. In order to characterize the connected graphs that achieve equality in the upper bound of Theorem 6.62, we first need a characterization of the 4-uniform hypergraphs that achieve equality in the bound of Theorem 6.63. The complement 𝐹 of a

182

Chapter 6. Upper Bounds in Terms of Minimum Degree

hypergraph 𝐹 is the hypergraph with the same vertex set 𝑉 (𝐹) and where 𝑒 is an edge in the complement 𝐹 if and only if 𝑉 (𝐹) \ 𝑒 is an edge in 𝐹, that is, 𝑒 ∈ 𝐸 (𝐹) if and only if 𝑉 (𝐹) \ 𝑒 ∈ 𝐸 (𝐹). If 𝐻 is the complement of the Fano plane, then 𝐻 is a 4-uniform hypergraph with 𝑛 = 7 vertices and 𝑚 = 7 edges satisfying 1 𝜏(𝐻) = 3 = 63 21 = 21 (5𝑛 + 4𝑚). With a more detailed analysis, Yeo [765] (see also [490]) showed that the complement of the Fano plane is the only hypergraph that achieves equality in the bound of Theorem 6.63. The well-known Heawood graph is shown in Figure 6.24(a). Theorem 6.64 ([765]) If 𝐻 is a 4-uniform hypergraph of order 𝑛 and size 𝑚, then 5 4 𝜏(𝐻) ≤ 21 𝑛 + 21 𝑚, with equality if and only if 𝐻 is the complement of the Fano plane. The bipartite complement of the Heawood graph, shown in Figure 6.24(b), is the bipartite graph formed by taking the two partite sets of the Heawood graph and joining a vertex from one partite set to a vertex from the other partite set by an edge whenever they are not joined in the Heawood graph. The bipartite complement of the Heawood graph can also be seen as the incidence bipartite graph of the complement of the Fano plane. The incidence bipartite graph 𝐺 of a hypergraph 𝐻 is the bipartite graph with partite sets (𝑋, 𝑌 ) where 𝑋 = 𝑉 (𝐻) and 𝑌 = 𝐸 (𝐻), where a vertex 𝑣 ∈ 𝑋 is joined to a vertex 𝑒 ∈ 𝑌 in the bipartite graph 𝐺 if the vertex 𝑣 ∈ 𝑉 (𝐻) is contained in the edge 𝑒 ∈ 𝐸 (𝐻) in the hypergraph 𝐻.

(a) The Heawood graph

(b) Bipartite complement of the Heawood graph

Figure 6.24 The Heawood graph and its bipartite complement Using the interplay between total domination in graphs and transversals in hypergraphs, as a consequence of Theorem 6.64, we have the following result, a proof of which is given in [490]. Theorem 6.65 ([490]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 4, then 𝛾t (𝐺) ≤ 37 𝑛, with equality if and only if 𝐺 is the bipartite complement of the Heawood graph.

6.3.6

Minimum Degree Five

Thomassé and Yeo [709] posed the following conjecture about the total domination number of a graph with minimum degree at least 5.

Section 6.3. Bounds on the Total Domination Number

183

Conjecture 6.66 (Thomassé-Yeo Conjecture [709]) If 𝐺 is a graph of order 𝑛 with 4 𝛿(𝐺) ≥ 5, then 𝛾t (𝐺) ≤ 11 𝑛. If Conjecture 6.66 is true, then the bound is best possible, as shown by Thomassé and Yeo [709], who constructed a 5-uniform hypergraph 𝐻 as follows. Let 𝐻 be the hypergraph with vertex set 𝑉 (𝐻) = [10] 0 and edge set 𝐸 (𝐻) = {𝑒 0 , 𝑒 1 , . . . , 𝑒 10 }, where the edge 𝑒 𝑖 = 𝑄 + 𝑖 for 𝑖 ∈ [10] 0 and where 𝑄 = {1, 3, 4, 5, 9} is the set of non-zero quadratic residues modulo 11. Thus, 𝑒 0 = {1, 3, 4, 5, 9}, 𝑒 1 = {2, 4, 5, 6, 10}, 𝑒 2 = {0, 3, 5, 6, 7}, . . . , 𝑒 10 = {0, 2, 3, 4, 8}. The hypergraph 𝐻 is a 5uniform, 5-regular hypergraph with 𝑛 = 11 vertices and 𝑚 = 11 edges, and with 4 transversal number 𝜏(𝐻) = 4 = 11 𝑛. Let 𝐺 22 be the incidence bipartite graph of the hypergraph 𝐻. The graph 𝐺 22 , illustrated in Figure 6.25, is a 5-regular (bipartite) 4 graph of order 𝑛 = 22 that satisfies 𝛾t (𝐺 22 ) = 8 = 11 𝑛.

Figure 6.25 A graph 𝐺 22 of order 𝑛 = 22 and 𝛾t (𝐺 22 ) = 8 =

4 11 𝑛

Several attempts were made to settle Conjecture 6.66, posed by Thomassé and Yeo. In 2015 Dorfling and Henning [243] established the following upper bound on the transversal number of a 5-uniform hypergraph in terms of its order and size. Theorem 6.67 ([243]) If 𝐻 is a 5-uniform hypergraph of order 𝑛 and size 𝑚, then 1 𝜏(𝐻) ≤ 44 (10𝑛 + 7𝑚). From Observation 6.56 and Theorem 6.67, one can prove the following upper bound on the total domination number of a graph with minimum degree at least 5. Theorem 6.68 ([243]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 5, then
4 1 𝛾t (𝐺) ≤ 17 44 𝑛 = 11 + 44 𝑛. The upper bound in Theorem 6.67 was subsequently improved in 2016 by Eustis et al. [263]. To state their result, for 𝑑 ≥ 0 an integer, let 𝑓 (𝑑) be the function defined as follows. For small 𝑑 ∈ [4] 0 , the values of 𝑓 (𝑑) are given in Table 6.9. For 𝑑 ≥ 5, let 𝑓 (𝑑) be defined recursively by 𝑓 (𝑑) = 𝑓 (𝑑 − 1) −

2 𝑓 (𝑑 − 1) − 𝑓 (𝑑 − 2) . 4𝑑

(6.12)

Chapter 6. Upper Bounds in Terms of Minimum Degree

184

𝑑

0

1

2

3

4

𝑓 (𝑑)

1

4 5

26 35

9 13

49 75

Table 6.9 Values of 𝑓 (𝑑) for small 𝑑 ∈ [4] 0 .

We are now in a position to state the result in [263]. Theorem 6.69 ([243]) If 𝐻 is a 5-uniform hypergraph of order 𝑛, then ∑︁
𝜏(𝐻) ≤ 𝑛 − 𝑓 deg 𝐻 (𝑣) . 𝑣 ∈𝑉 (𝐻 )

We remark that the proof of Theorem 6.69 given in [243] is constructive, and a transversal 𝑇
in a 5-uniform hypergraph of order 𝑛 satisfying |𝑇 | ≤ 𝑛 − Í 𝑣 ∈𝑉 (𝐻 ) 𝑓 deg 𝐻 (𝑣) can be found in polynomial time. As shown in [243], the function 𝑓 defined in Equation (6.12) is convex, assuming we extend the domain of 𝑓 to the nonnegative reals by linear interpolation. This implies that if 𝑑1 , 𝑑2 , . . . , 𝑑 𝑛 are arbitrary nonnegative reals, then  ∑︁  𝑛 𝑛 ∑︁ 1 1 𝑓 (𝑑𝑖 ) ≥ 𝑓 𝑛 𝑑𝑖 . 𝑛 𝑖=1

𝑖=1

Hence, if the 5-uniform hypergraph 𝐻 of order 𝑛 has size 𝑚 = 𝑛, then ∑︁ 𝑛 = 𝑚 = 15 deg 𝐻 (𝑣), 𝑣 ∈𝑉 (𝐻 )

implying that ∑︁




𝑓 deg 𝐻 (𝑣) ≥ 𝑛 · 𝑓

𝑣 ∈𝑉 (𝐻 )



∑︁

1 deg 𝐻 (𝑣) 𝑛 𝑣 ∈𝑉 (𝐻 )

 = 𝑛 · 𝑓 (5).

Therefore, as a consequence of Theorem 6.69, we have the following result. Theorem 6.70 ([243]) If 𝐻 is a 5-uniform hypergraph of order 𝑛 and size 𝑛, then
2453 𝜏(𝐻) ≤ 1 − 𝑓 (5) 𝑛 = 𝑛 ≈ 0.3773 𝑛. 6500 From Observation 6.56 and Theorem 6.70, one can prove the following upper bound on the total domination number of a graph with minimum degree at least 5. Theorem 6.71 ([243]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 5, then   2453 4 11 𝛾t (𝐺) ≤ 𝑛< + 𝑛. 6500 11 800 However, Conjecture 6.66 has yet to be settled. To achieve this goal, a new idea different than those employed in the proofs of Theorems 6.67 and 6.70 seems necessary.

Section 6.3. Bounds on the Total Domination Number

185

6.3.7 Minimum Degree Six It is a natural question to ask whether the Thomassé-Yeo Conjecture 6.66 holds if we relax the minimum degree condition in the statement of the conjecture from minimum degree at least 5 to minimum degree at least 6. This is indeed the case, as shown by Henning and Yeo [493]. In order to state their result, we define the weight w(𝑒) of an edge 𝑒 in a 6-uniform hypergraph 𝐻 as w(𝑒) = 7707. Further, we define the weight of the vertex 𝑣, denoted w(𝑣), in a 6-uniform hypergraph 𝐻 as 7707 if deg 𝐻 (𝑣) ≥ 4, and we define w(𝑣) as 7524, 6861, and 5788 if deg 𝐻 (𝑣) = 𝑖 and 𝑖 equals 3, 2, and 1, respectively. Thus, the weight w(𝑣) of a vertex 𝑣 in the hypergraph 𝐻 is given in Table 6.10. deg 𝐻 (𝑣)

≥4

3

2

1

0

w(𝑣)

7707

7524

6861

5788

0

Table 6.10 The weight w(𝑣) assigned to a vertex 𝑣 in the hypergraph 𝐻 We define the weight of the 6-uniform hypergraph 𝐻 of order 𝑛 and size 𝑚 as ∑︁ ∑︁ w(𝐻) = w(𝑒) + w(𝑣). 𝑒∈𝐸 (𝐻 )

𝑣 ∈𝑉 (𝐻 )

Equivalently, w(𝐻) = 7707𝑛 ≥4 + 7524𝑛3 + 6861𝑛2 + 5788𝑛1 + 7707𝑚,

(6.13)

where 𝑛𝑖 for 𝑖 ∈ [3] 0 denotes the number of vertices of degree 𝑖 in 𝐻, and 𝑛 ≥4 denotes the number of vertices of degree at least 4 in 𝐻. Thus, 𝑛 = 𝑛 ≥4 + 𝑛3 + 𝑛2 + 𝑛1 + 𝑛0 . We are now in a position to state the result given in [493]. Theorem 6.72 ([493]) If 𝐻 is a 6-uniform hypergraph, then 42435𝜏(𝐻) ≤ w(𝐻). As a consequence of Theorem 6.72, we have the following result. Corollary 6.73 ([493]) If 𝐻 is a 6-uniform hypergraph of order 𝑛 and size 𝑚, then 𝜏(𝐻) ≤

2569 (𝑛 + 𝑚). 14145

Proof Let 𝐻 be a 6-uniform hypergraph. By Equation (6.13), w(𝐻) = 5788𝑛1 + 6861𝑛2 + 7524𝑛3 + 7707𝑛 ≥4 + 7707𝑚 ≤ 7707(𝑛0 + 𝑛1 + 𝑛2 + 𝑛3 + 𝑛 ≥4 ) + 7707𝑚 = 7707(𝑛 + 𝑚).

Chapter 6. Upper Bounds in Terms of Minimum Degree

186 Thus, by Theorem 6.72,

7707(𝑛 + 𝑚) ≥ w(𝐻) ≥ 42435𝜏(𝐻), or equivalently, 𝜏(𝐻) ≤

2569 (𝑛 + 𝑚). 14145

As an immediate consequence of Observation 6.56 and Corollary 6.73, we have the following upper bound on the total domination number of a graph with minimum degree at least 6. Theorem 6.74 ([493]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then 𝛾t (𝐺) ≤

5138 𝑛 ≈ 0.3773 𝑛. 14145

Proof Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺) ≥ 6. Let 𝐻 be the open neighborhood hypergraph ONH(𝐺) of 𝐺. Thus, 𝐻 has order 𝑛(𝐻) = 𝑛 and size 𝑚(𝐻) = 𝑛, where all edges have size at least 𝛿(𝐺) ≥ 6. Hence, by Observation 6.56 and Theorem 6.74, 𝛾t (𝐺) = 𝜏(ONH(𝐺)) ≤


2569 2569 5138 𝑛(𝐻) + 𝑚(𝐻) = (𝑛 + 𝑛) = 𝑛, 14145 14145 14145

completing the proof of the theorem. 5138 4 1 We note that 14145 . Hence, as an immediate consequence of Theo< 11 − 2510 rem 6.74, the Thomassé-Yeo Conjecture 6.66 holds if the minimum degree is at least 6. The upper bound in Theorem 6.74 was slightly improved by the same authors in [495] as an application of the so-called Tuza-Vestergaard Theorem, which states that the transversal number of a 6-uniform hypergraph of order 𝑛 is at most 𝑛4 .

Theorem 6.75 ([495]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then 𝛾t (𝐺) ≤ 4549 13299 𝑛 ≈ 0.3420 𝑛. The bound in Theorem 6.75 is currently the best known upper bound on the total domination number of a graph with minimum degree at least 6. However, it is not known if there is a graph with minimum degree at least 6 that achieves this bound. It is unlikely that this bound is achievable. Henning and Yeo [493] pose the following conjecture. Conjecture 6.76 ([493]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then 𝛾t (𝐺) ≤ 4 13 𝑛. If Conjecture 6.76 is true, then the bound is best possible as shown by the authors in [493], who construct a 5-uniform hypergraph 𝐻 as follows. Let 𝐻 be the hypergraph with vertex set 𝑉 (𝐻) = [12] 0 and edge set 𝐸 (𝐻) = {𝑒 0 , 𝑒 1 , . . . , 𝑒 12 }, where the edge 𝑒 𝑖 = 𝑄 + 𝑖 for 𝑖 ∈ [12] 0 and where 𝑄 is the set of non-zero quadratic residues modulo 13, that is, 𝑄 = {1, 3, 4, 9, 10, 12}. The hypergraph 𝐻 is a 6-regular, 6-uniform hypergraph with 𝑛(𝐻) = 13 vertices
and 𝑚(𝐻) = 13 edges, and with 2 transversal number 𝜏(𝐻) = 4 = 13 𝑛(𝐻) + 𝑚(𝐻) .

Section 6.3. Bounds on the Total Domination Number

187

Let 𝐺 26 be the incidence bipartite graph of the hypergraph 𝐻. Thus, 𝐺 26 is the bipartite graph with partite sets (𝑋, 𝑌 ) where 𝑋 = 𝑉 (𝐻) and 𝑌 = 𝐸 (𝐻), where a vertex 𝑣 ∈ 𝑋 is joined to a vertex 𝑒 ∈ 𝑌 in the bipartite graph 𝐺 if the vertex 𝑣 ∈ 𝑉 (𝐻) is contained in the edge 𝑒 ∈ 𝐸 (𝐻) in the hypergraph 𝐻. The graph 𝐺 26 , shown in Figure 6.26, is a 6-regular (bipartite) graph of order 𝑛 = 26 satisfying 4 𝛾t (𝐺 26 ) = 8 = 13 𝑛. Thus, if 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 6, then the best 4 upper bound we can hope for is 𝛾t (𝐺) ≤ 13 𝑛. Conjecture 6.76 claims that this is the best possible upper bound.

Figure 6.26 A graph 𝐺 26 of order 𝑛 = 26 satisfying 𝛾t (𝐺 22 ) = 8 =

4 13 𝑛

We summarize the best known upper bounds on the total domination number of a graph 𝐺 in terms of its order 𝑛 and minimum degree 𝛿 ∈ [6] in Table 6.11. The bounds for minimum degree 𝛿 ∈ [3] are tight (in the sense that there are infinitely many connected graphs that achieve the bound in each case). However, although the bound for minimum degree 𝛿 = 4 is best possible, it is not known if the bound is tight. As remarked earlier, the known bounds for minimum degree 𝛿 = 5 and 4 𝛿 = 6 are unlikely best possible, and the conjectured bounds are 𝛾t (𝐺) ≤ 11 𝑛 and 4 𝛾t (𝐺) ≤ 13 𝑛, respectively.

6.3.8

A Heuristic Bound

In this section, we present a heuristic algorithm that yields an upper bound on the total domination of a graph in terms of its order and minimum degree. Moreover, we show that this simple greedy approximation algorithm can be used to efficiently find a TD-set whose cardinality is “close” to the cardinality of a minimum TD-set. We shall prove the following 2007 result due to Henning and Yeo [482]. Theorem 6.77 ([482]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) = 𝛿 ≥ 1, then   1 + ln(𝛿) 𝛾t (𝐺) ≤ 𝑛. 𝛿

Chapter 6. Upper Bounds in Terms of Minimum Degree

188 Year

𝛿≥

1980

𝛿(𝐺) ≥ 1

𝛾t (𝐺) ≤

2000

𝛿(𝐺) ≥ 2

𝛾t (𝐺) ≤

2004

𝛿(𝐺) ≥ 3

𝛾t (𝐺) ≤

2007

𝛿(𝐺) ≥ 4

𝛾t (𝐺) ≤

2016

𝛿(𝐺) ≥ 5

𝛾t (𝐺) ≤

2022

𝛿(𝐺) ≥ 6

𝛾t (𝐺) ≤

a b

𝛾t ≤

Citation

2 a 3𝑛 4 b 7𝑛 1 2𝑛 3 7𝑛
2453 4 11 6500 𝑛 < 11 + 800 𝑛
4549 4 17 13299 𝑛 < 13 + 494 𝑛

[182] [453] [35, 176, 720] [709] [263] [495]

If 𝑛 ≥ 3 and 𝐺 is connected. If 𝑛 ≥ 11 and 𝐺 is connected. Table 6.11 Upper bounds on 𝛾t (𝐺) with minimum degree 𝛿 ∈ [6]

Further, using a greedy algorithm we can, in time complexity O (𝑛 + 𝛿 𝑛), find a 𝛿)
TD-set 𝑇 in the graph 𝐺 such that |𝑇 | ≤ 1+ln( 𝑛. 𝛿 Proof Let 𝐺 be a graph with minimum degree 𝛿 ≥ 1 and order 𝑛. If 𝛿 = 1, 𝛿)
then 𝛾t (𝐺) ≤ 𝑛 = 1+ln( 𝑛. Hence, we may assume that 𝛿 ≥ 2, for otherwise 𝛿 the result is immediate. We now consider the hypergraph 𝐻 obtained from the open neighborhood hypergraph ONH(𝐺) by shrinking all edges of ONH(𝐺), if necessary, to edges of size 𝛿. By construction, 𝐻 is a 𝛿-uniform hypergraph with 𝑛 vertices and 𝑛 edges. Every transversal in 𝐻 is a transversal in ONH(𝐺) and so, 𝛾t (𝐺) = 𝜏(ONH(𝐺)) ≤ 𝜏(𝐻). We now greedily select vertices of maximum degree at every stage in the algorithm until we obtain a transversal in 𝐻. The resulting set is a TD-set of 𝐺. More precisely, we construct a transversal 𝑇 in 𝐻 as follows. Initially, we select any vertex 𝑣 of maximum degree in 𝐻, delete 𝑣 and all edges incident with 𝑣 from 𝐻, and let 𝑇 = {𝑣}. We note that the resulting hypergraph 𝐻 − 𝑣 is a 𝛿-uniform hypergraph with at most 𝑛 vertices. In this resulting hypergraph, we select any vertex 𝑤 of maximum degree, delete 𝑤 and all edges incident with it, and then add 𝑤 to the set 𝑇. We continue this process until no edges remain. By construction, every deleted edge contains at least one vertex of the resulting set 𝑇, implying that the set 𝑇 is a transversal in 𝐻. Hence, 𝜏(𝐻) ≤ |𝑇 |.
𝛿) We show next that |𝑇 | ≤ 1+ln( 𝑛. Let 𝐻 𝛿 = 𝐻 and let 𝑇 𝛿 denote the set of 𝛿 vertices in 𝑇 of degree 𝛿 or more in the current hypergraph when they were added to the transversal 𝑇. Let 𝑡 𝛿 = |𝑇 𝛿 |. For 𝑗 ∈ [𝛿 − 1], let 𝑇 𝑗 be the set of all vertices that had degree exactly 𝑗 in the current hypergraph when they were added to the transversal 𝑇, and let 𝑡 𝑗 = |𝑇 𝑗 |. By definition, 𝑇=

𝛿 Ø 𝑗=1

𝑇𝑖

and

|𝑇 | =

𝛿 ∑︁ 𝑗=1

𝑡𝑗.

Section 6.3. Bounds on the Total Domination Number

189

For 𝑗 ∈ [𝛿 − 1], let 𝐻 𝑗 = 𝐻 𝑗+1 − 𝑇 𝑗+1 and so, 𝐻 𝑗 is obtained from the hypergraph 𝐻 𝑗+1 by deleting all vertices in 𝑇 𝑗+1 and all edges incident with vertices in 𝑇 𝑗+1 . We note that 𝐻 𝑗 is a 𝛿-uniform hypergraph with at most 𝑛 vertices and with maximum degree Δ(𝐻 𝑗 ) ≤ 𝑗, implying that ∑︁ deg 𝐻 𝑗 (𝑣) ≤ 𝑗𝑛. (6.14) 𝑣 ∈𝑉 (𝐻 𝑗 )

For 𝑗 ∈ [𝛿 − 1] \ {1}, the hypergraph 𝐻 𝑗 −1 is obtained from 𝐻 𝑗 by deleting exactly 𝑗𝑡 𝑗 edges and so, |𝐸 (𝐻 𝑗 )| = |𝐸 (𝐻 𝑗 −1 )| + 𝑗𝑡 𝑗 .

(6.15)

Since Δ(𝐻1 ) ≤ 1, the edges of 𝐻1 have no vertices in common and so, by construction, exactly one vertex from each edge of 𝐻1 is added to the set 𝑇1 . Thus, |𝐸 (𝐻1 )| = 𝑡 1 . For each value of 𝑗 with 𝑗 ∈ [𝛿 − 1], it therefore follows from Inequality (6.14) and Equation (6.15) that 𝑗 ∑︁

𝑖𝑡𝑖 = 𝑡1 +

𝑖=1

𝑗 ∑︁

𝑖𝑡𝑖 = 𝑡1 +

𝑖=2

𝑗 ∑︁

|𝐸 (𝐻𝑖 )| − |𝐸 (𝐻𝑖−1 )|




𝑖=2

= |𝐸 (𝐻 𝑗 )| ∑︁ = 1𝛿 deg 𝐻 𝑗 (𝑣) 𝑣 ∈𝑉 (𝐻 𝑗 )

≤

𝑗𝑛 𝛿 .

The hypergraph 𝐻 𝛿−1 is obtained from 𝐻 𝛿 by deleting at least 𝛿𝑡 𝛿 edges and so, |𝐸 (𝐻 𝛿 )| ≥ |𝐸 (𝐻 𝛿−1 )| + 𝛿𝑡 𝛿 .

(6.16)

Recall that since 𝐻 𝛿 = 𝐻, the hypergraph 𝐻 𝛿 has exactly 𝑛 edges. Thus, |𝐸 (𝐻 𝛿 )| = 𝑛, implying by Inequality (6.16) that 𝛿 ∑︁ 𝑖=1

𝑖𝑡𝑖 = 𝛿𝑡 𝛿 +

𝛿−1 ∑︁

𝑖𝑡𝑖 = 𝛿𝑡 𝛿 + |𝐸 (𝐻 𝛿−1 )|

𝑖=1


≤ |𝐸 (𝐻 𝛿 )| − |𝐸 (𝐻 𝛿−1 )| + |𝐸 (𝐻 𝛿−1 )| = |𝐸 (𝐻 𝛿 )| = 𝑛. Thus, for each value of 𝑗 ∈ [𝛿], the inequality 𝑗 ∑︁ 𝑖=1

𝑖𝑡𝑖 ≤

𝑗𝑛 𝛿

Chapter 6. Upper Bounds in Terms of Minimum Degree

190

holds. Therefore, for each value of 𝑗 ∈ [𝛿] 0 , 𝑗 ∑︁

𝑖𝑡𝑖 =

𝑗𝑛 − 𝜀𝑗 𝛿

(6.17)

𝑖=1

for some real number 𝜀 𝑗 ≥ 0, where 𝜀0 = 0. By Equation (6.17), for each value of 𝑗 ∈ [𝛿], 𝑗 −1 ∑︁

𝑗 ∑︁



   𝑗𝑛 ( 𝑗 − 1)𝑛 𝑛 𝑗𝑡 𝑗 = 𝑖𝑡𝑖 − 𝑖𝑡𝑖 = − 𝜀𝑗 − − 𝜀 𝑗 −1 = − (𝜀 𝑗 − 𝜀 𝑗 −1 ), 𝛿 𝛿 𝛿 𝑖=1 𝑖=1 or equivalently, 𝑡𝑗 =

𝜀 𝑗 − 𝜀 𝑗 −1 𝑛 − . 𝑗𝛿 𝑗

(6.18)

Thus, by Equation (6.18), |𝑇 | =

𝛿 ∑︁

𝑡𝑗 =

𝑗=1

 𝛿  ∑︁ 𝜀 𝑗 − 𝜀 𝑗 −1 𝑛 − 𝑗𝛿 𝑗 𝑗=1

=

   𝛿  𝛿−1 ∑︁ 𝜀 𝛿 ∑︁ 𝑛 1 1 − − 𝜀𝑗 − 𝑗𝛿 𝛿 𝑗 𝑗 +1 𝑗=1 𝑗=1

≤

 𝛿  ∑︁ 𝑛 𝑗𝛿 𝑗=1

=

𝑛 ∑︁ 1 𝛿 𝑗=1 𝑗

≤


𝑛 1 + ln(𝛿) . 𝛿

𝛿

Therefore, by our earlier observations,  𝛾t (𝐺) = 𝜏(ONH(𝐺)) ≤ 𝜏(𝐻) ≤ |𝑇 | ≤

 1 + ln(𝛿) 𝑛. 𝛿

This establishes the desired upper bound. We next give a brief discussion of the complexity of the greedy algorithm. With correct implementation, the greedy algorithm to build the set 𝑇 can be made to run in time O (𝑛 + 𝛿 𝑛). It would require us to keep a data structure, such as a hash table, that enables one to find a vertex of maximum degree in constant time. Once such a vertex is deleted from the hypergraph and added to the set 𝑇, the degrees of the vertices in the resulting hypergraph need to be updated. Since the sum of the degrees of vertices in 𝐻 is at most ∑︁ deg 𝐻 (𝑣) = 𝛿 · 𝑚(𝐻) = 𝛿 · 𝑛, 𝑣 ∈𝑉 (𝐻 )

Section 6.4. Bounds on the Independent Domination Number

191

we need to update the degrees at most 𝛿 𝑛 times. The overall complexity of the greedy algorithm can be shown to be O (𝑛 + 𝛿 · 𝑛). It can be deduced from a probabilistic proof due to Alon [17] that the bound in Theorem 6.77 is asymptotically (that is, when 𝛿 → ∞) optimal. We remark that this can also be deduced from the following 2007 result due to Thomassé and Yeo [709]. Theorem 6.78 ([709]) For any 𝜀 > 0 and for sufficiently large 𝛿, there exists a 𝛿-uniform hypergraph 𝐻 of order 𝑛 and size 𝑛 edges satisfying   (1 − 𝜀) ln(𝛿) 𝜏(𝐻) > 𝑛. 𝛿 Thomassé and Yeo [709] also provided the following result showing that the bound on the TD-set produced by the greedy algorithm in the proof of Theorem 6.77 is close to optimal. Theorem 6.79 ([709]) For every integer 𝛿 ≥ 1, there exists a bipartite 𝛿-regular graph 𝐺 of order 𝑛 satisfying   0.1 ln(𝛿) 𝛾t (𝐺) > 𝑛. 𝛿 Although the bound on the total domination number constructed by the heuristic (greedy) algorithm presented in the proof of Theorem 6.77 is asymptotically optimal, the bound is far from optimal when the minimum degree 𝛿 is small. However, for 𝛿 ≥ 8, the upper bound in Theorem 6.77 is better than the upper bound given in Theorem 6.60. In Table 6.12 we compare these two bounds for small 𝛿 ∈ {5, 6, 7, 8, 9}. 𝛿 1 + ln(𝛿) 𝛿   1 + 2𝛿  3𝛿 

5

6

7

8

9

≈ 0.5218

≈ 0.4652

≈ 0.4208

≈ 0.3849

≈ 0.3552

≈ 0.4285

≈ 0.4444

0.4

≈ 0.4166

≈ 0.3846

2

Table 6.12 Comparing the bounds of Theorems 6.60 and 6.77 for 𝛿 ∈ {5, 6, 7, 8, 9} Thus, for example if 𝐺 is a graph of order 𝑛 and 𝛿(𝐺) ≥ 8, then 𝛾t (𝐺) ≤ 0.4166 𝑛 by Theorem 6.60 and 𝛾t (𝐺) ≤ 0.3849 𝑛 by Theorem 6.77.

6.4

Bounds on the Independent Domination Number

In this section, we present a breakthrough result by Sun and Wang that if 𝐺 is a graph of order 𝑛 with minimum degree 𝛿(𝐺) = 𝛿 where 𝛿 ≥ 1 is an arbitrary integer, then

192

Chapter 6. Upper Bounds in Terms of Minimum Degree

√ the independent domination number 𝑖(𝐺) is bounded above by 𝑛 + 2𝛿 − 2 𝛿𝑛, thereby proving a conjecture posed by Favaron. We also present a classical result of Rosenfeld that the independence number 𝛼(𝐺) of a regular graph 𝐺 is at most one-half its order. We show that equality in Rosenfeld’s bound for the independent domination number 𝑖(𝐺) is only obtainable for graphs with every component a balanced complete bipartite graph.

6.4.1

Minimum Degree One

As in previous sections, we repeat the upper bounds from Chapter 4 for the case of minimum degree one. Observation 6.80 ([398]) If 𝐺 is a graph of order 𝑛, then 𝑖(𝐺) ≤ 𝑛 − Δ(𝐺). Theorem 6.81 ([84]) If 𝐺 is an isolate-free graph of order 𝑛, then 𝑖(𝐺) ≤ 𝑛 + 2 − 𝛾(𝐺) −

𝑛 . 𝛾(𝐺)

Theorem 6.82 ([274]) If 𝐺 is an isolate-free graph of order 𝑛, then √ 𝑖(𝐺) ≤ 𝑛 + 2 − 2 𝑛.

6.4.2

Arbitrary Minimum Degree

In this section, we present a tight upper bound on the independent domination number of a graph with arbitrary minimum degree. In 1991 Haviland [398] improved the bound of Favaron in Theorem 6.82 when 𝛿 ≥ 14 𝑛. In order to state this result, we first present the following two lemmas of Haviland [398]. Recall that if 𝑣 is a vertex of 𝐺 and 𝑋 ⊆ 𝑉, then the degree of 𝑣 in 𝑋, denoted deg𝑋 (𝑣), is the number of neighbors of 𝑣 in 𝐺 that belong to the set 𝑋. Lemma 6.83 ([398]) If there exists an 𝑖-set 𝐼 of a graph 𝐺 of order 𝑛 with√𝛿(𝐺) = 𝛿 such that no vertex outside 𝐼 dominates all vertices in 𝐼, then 𝑖(𝐺) ≤ 𝑛 − 𝛿𝑛. Proof Let 𝐼 be an 𝑖-set of 𝐺 such that no vertex in 𝐼 = 𝑉 \ 𝐼 dominates all vertices in 𝐼. Among all vertices in 𝐼, let 𝑥 be chosen to have the maximum number of neighbors in 𝐼, that is, deg𝐼 (𝑥) is a maximum. Let 𝐼 𝑥 = 𝐼 ∩ N𝐺 (𝑥). Let 𝑋 = {𝑣 ∈ 𝐼 : N𝐺 (𝑣) ⊆ 𝐼 𝑥 } and let 𝑊 be a maximal independent set in 𝐺 [𝑋] containing the vertex 𝑥. Since the set 𝑊 ∪ (𝐼 \ 𝐼 𝑥 ) is a maximal independent set, |𝑊 | + |𝐼 | − |𝐼 𝑥 | ≥ 𝑖(𝐺) = |𝐼 |, implying that |𝑊 | ≥ |𝐼 𝑥 |. The number of edges between 𝐼 and 𝐼 is at least 𝛿|𝐼 |, noting that every vertex in 𝐼 has at least 𝛿 neighbors in 𝐼. By the Pigeonhole Principle, there is a vertex in 𝐼 with at least 𝛿|𝐼 |/|𝐼 | = 𝛿|𝐼 |/(𝑛 − |𝐼 |) neighbors in 𝑋. Hence, by our choice of the vertex 𝑥, 𝛿 · |𝐼 | |𝑋 | ≥ |𝑊 | ≥ |𝐼 𝑥 | ≥ . (6.19) 𝑛 − |𝐼 |

Section 6.4. Bounds on the Independent Domination Number

193

By supposition, |𝐼 \ 𝐼 𝑥 | ≥ 1. By definition of the set 𝑋, each vertex in 𝐼 \ 𝐼 𝑥 has all its neighbors in 𝐼 \ 𝑋, implying that |𝐼 \ 𝑋 | ≥ 𝛿.

(6.20)

By Inequalities (6.19) and (6.20), 𝑛 − |𝐼 | = |𝑋 | + |𝐼 \ 𝑋 | ≥

𝛿 · |𝐼 | + 𝛿, 𝑛 − |𝐼 |

or equivalently, |𝐼 | 2 − 2𝑛|𝐼 | − 𝑛(𝑛 − 𝛿) ≥ 0.

(6.21)

√ Solving the quadratic expression in Inequality (6.21) yields 𝑖(𝐺) = |𝐼 | ≤ 𝑛− 𝛿𝑛. Lemma 6.84 ([398]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 14 𝑛, then 𝑖(𝐺) ≤ 12 𝑛. Proof Let 𝐼 be an 𝑖-set of 𝐺. If Δ(𝐺) ≥ 𝑖(𝐺), then by Observation 6.80, 𝑖(𝐺) ≤ 𝑛 − Δ(𝐺) ≤ 𝑛 − 𝑖(𝐺) and so, 𝑖(𝐺) ≤ 12 𝑛. Hence, we may assume that Δ(𝐺) < 𝑖(𝐺), implying that no vertex outside 𝐼 dominates all vertices in 𝐼. Therefore, we can apply Lemma 6.83 to yield √︃ √ 𝑖(𝐺) ≤ 𝑛 − 𝛿𝑛 ≤ 𝑛 − 14 𝑛2 = 12 𝑛. We now present Haviland’s improvement of the bound due to Favaron in Theorem 6.82 when 𝛿 is large relative to the order. Theorem 6.85 ([398]) If 𝐺 is a graph of order 𝑛,
then the following hold: (a) If 14 𝑛 ≤ 𝛿(𝐺) ≤ 25 𝑛, then 𝑖(𝐺) ≤ 23 𝑛 − 𝛿(𝐺) . (b) If 25 𝑛 ≤ 𝛿(𝐺) ≤ 12 𝑛, then 𝑖(𝐺) ≤ 𝛿(𝐺). Proof degree 𝛿(𝐺) = 𝛿. We note that √ Let 𝐺 be a graph of order 𝑛 with minimum √ 𝑛 − 𝛿𝑛 ≤ 23 (𝑛 − 𝛿) if and only if 𝑛 + 2𝛿 ≤ 3 𝛿𝑛 if and only if 4𝛿2 − 5𝛿𝑛 + 𝑛2 ≤ 0 if and only if 14 𝑛 ≤ 𝛿 ≤ 𝑛. Further, we note that 23 (𝑛 − 𝛿) ≤ 𝛿 if and only if 25 𝑛 ≤ 𝛿 ≤ 𝑛. √  Thus, if 𝑖(𝐺) ≤ max 𝛿, 𝑛 − 𝛿𝑛 , then our desired bounds in parts (a) and (b) of the √  theorem hold. Hence, we may assume that 𝑖(𝐺) > max 𝛿, 𝑛 − 𝛿𝑛 . Let 𝐼 be an 𝑖-set √ of 𝐺, and let 𝐼 = 𝑉 \ 𝐼. By Lemma 6.83 and by our assumption that 𝑖(𝐺) > 𝑛 − 𝛿𝑛, there exists a vertex in 𝐼 that dominates all vertices in 𝐼. Let 𝑆 be the set of all such vertices and so,  𝑆 = 𝑣 ∈ 𝐼 : 𝐼 ⊆ N𝐺 (𝑣) . Suppose that 𝑆 = 𝐼, implying that all edges between 𝐼 and 𝐼 are present. Thus, every vertex in 𝐼 has degree at least |𝐼 | = 𝑖(𝐺) > 𝛿, and every vertex in 𝐼 has degree exactly |𝐼 | = 𝑛 − |𝐼 |. Since 𝐺 has minimum degree 𝛿, this implies that 𝑖(𝐺) > 𝛿 = |𝐼 | = 𝑛 − 𝑖(𝐺) and so, 𝑖(𝐺) > 12 𝑛. However, every maximal independent set in 𝐺 [𝑆] is an ID-set of 𝐺 and so, 𝑖(𝐺) ≤ |𝑆| = 𝑛 − 𝑖(𝐺). Therefore, 𝑖(𝐺) ≤ 12 𝑛, a contradiction. Hence, 𝑆 is a proper subset of 𝐼. Let  𝑇 = 𝑣 ∈ 𝐼 \ 𝑆 : 𝑆 ⊆ N𝐺 (𝑣)

Chapter 6. Upper Bounds in Terms of Minimum Degree

194

and so, 𝑇 ⊆ 𝐼 \ 𝑆. Suppose that 𝑇 = 𝐼 \ 𝑆. In this case, every maximal independent set in 𝐺 [𝑆] is an ID-set of 𝐺 and so, |𝑆| ≥ 𝑖(𝐺) > 𝛿, implying that every vertex in 𝐼 ∪ 𝑇 has degree greater than 𝛿 in 𝐺. Further, every vertex in 𝑆 has degree at least |𝐼 | = 𝑖(𝐺) > 𝛿. Hence, all vertices of 𝐺 have degree greater than 𝛿, a contradiction. Thus, 𝑇 is a proper subset of 𝐼 \ 𝑆. Suppose that there exists a vertex 𝑣 ∈ 𝐼 \ (𝑆 ∪ 𝑇) that has more than 𝑛 − 2|𝐼 | neighbors in 𝐼, that is, deg𝐼 (𝑣) = |N𝐺 (𝑣) ∩ 𝐼 | > 𝑛 − 2|𝐼 | = 𝑛 − 2 𝑖(𝐺). By definition, there is a vertex 𝑢 ∈ 𝑆 that is not adjacent to the vertex 𝑣. Let 𝐼𝑢,𝑣 be a maximal independent set in 𝐺 that contains the nonadjacent pair {𝑢, 𝑣}. Since the vertex 𝑢 dominates the set 𝐼, we note that the ID-set 𝐼𝑢,𝑣 contains no vertex from the set 𝐼. Further, the ID-set 𝐼𝑢,𝑣 contains no neighbor of 𝑣, implying that 𝑖(𝐺) ≤ |𝐼𝑢,𝑣 | ≤
𝑛 − |𝐼 | − deg𝐼 (𝑣) < 𝑛 − 𝑖(𝐺) − 𝑛 − 2 𝑖(𝐺) = 𝑖(𝐺), a contradiction. Hence, for every vertex 𝑣 ∈ 𝐼 \ (𝑆 ∪ 𝑇), deg𝐼 (𝑣) ≤ 𝑛 − 2|𝐼 |. (6.22) Let 𝑥 ∈ 𝐼 \ (𝑆 ∪ 𝑇). Further, let 𝐼 𝑥 = 𝐼 ∩ N𝐺 (𝑥). By Inequality (6.22), we have 𝛿 ≤ deg𝐺 (𝑥) = deg𝐼 (𝑥) + deg𝐼 (𝑥) ≤ |𝐼 𝑥 | + 𝑛 − 2|𝐼 | and so, |𝐼 𝑥 | ≥ 𝛿 − 𝑛 + 2|𝐼 |.

(6.23)

Let 𝑋 = {𝑣 ∈ 𝐼 : N𝐺 (𝑣) ∩ 𝐼 ⊆ 𝐼 𝑥 } and let 𝑅 be a maximal independent set of 𝐺 [𝑋] that contains the vertex 𝑣. The set 𝑅 ∪ (𝐼 \ 𝐼 𝑥 ) is a maximal independent set of 𝐺, implying that |𝑅| + |𝐼 | − |𝐼 𝑥 | ≥ 𝑖(𝐺) = |𝐼 | and so, |𝑋 | ≥ |𝑅| ≥ |𝐼 𝑥 |. Thus, by Inequality (6.23), |𝑋 | ≥ 𝛿 − 𝑛 + 2|𝐼 |. (6.24) Since 𝑥 ∉ 𝑆, we know that 𝐼 \ 𝐼 𝑥 ≠ ∅. Further, N𝐺 (𝐼 \ 𝐼 𝑥 ) ⊆ 𝐼 \ 𝑋 and each vertex in 𝐼 \ 𝐼 𝑥 has at least 𝛿 neighbors in 𝐼 \ 𝑋. Thus, 𝑛 − |𝐼 | − |𝑋 | = |𝐼 | − |𝑋 | = |𝐼 \ 𝑋 | ≥ 𝛿.

(6.25)

By Inequalities (6.24) and (6.25), 𝑛 − |𝐼 | ≥ |𝑋 | + 𝛿 ≥ 2𝛿 − 𝑛 + 2|𝐼 | and so, 𝑖(𝐺) = |𝐼 | ≤ 23 (𝑛 − 𝛿). This proves (a). For 𝛿 ≥ 25 𝑛, we note that 23 (𝑛 − 𝛿) ≤ 𝛿, which completes the proof of part (b) and completes the proof of Theorem 6.85. We remark that if 25 𝑛 ≤ 𝛿 ≤ 12 𝑛, then the complete bipartite graph 𝐾 𝛿,𝑛− 𝛿 satisfies 𝑖(𝐺) = 𝛿 and so, the bound in Theorem 6.85(b) is best possible. However, the bound in Theorem 6.85(a) can be improved. Given a connected graph 𝐺 with arbitrary minimum degree 𝛿 ≥ 4, a tight upper bound (that holds for connected graphs of arbitrarily large order) on 𝛾(𝐺) has yet to be determined, even for the special case when 𝛿 = 4. However, this is not the case

Section 6.4. Bounds on the Independent Domination Number

195

for the independent domination number. In 1988 Favaron [274] conjectured such an upper bound on 𝑖(𝐺) as a function of 𝑛 and 𝛿. Conjecture 6.86 ([274]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) = 𝛿, then √ 𝑖(𝐺) ≤ 𝑛 + 2𝛿 − 2 𝛿𝑛. If 𝛿 = 0, then Conjecture 6.86 simplifies to 𝑖(𝐺) ≤ 𝑛, which is obviously true. If 𝛿 = 1, then Conjecture 6.86 is precisely the statement of Theorem 6.82, which was proven by Favaron. Hence, at the time Favaron posed Conjecture 6.86, it was only of interest for 𝛿 ≥ 2. Favaron [274] showed that for every positive integer 𝛿, the bound in Conjecture 6.86 is attained for infinitely many graphs as follows. For 𝛿 ≥ 1 and ℓ ≥ 2, let 𝐵𝑖 be the complete bipartite graph 𝐾 𝛿, 𝛿 (ℓ −1) with partite sets 𝑋𝑖 and 𝑌𝑖 where |𝑋𝑖 | = 𝛿 and |𝑌𝑖 | = 𝛿(ℓ − 1) for 𝑖 ∈ [ℓ]. Let 𝐺 𝛿,ℓ be the graph obtained from the disjoint union of the graphs 𝐵1 , 𝐵2 , . . . , 𝐵ℓ by adding all edges between the sets 𝑋𝑖 and 𝑋 𝑗 for all 𝑖 and 𝑗, where 𝑖, 𝑗 ∈ [ℓ] and 𝑖 ≠ 𝑗. The resulting graph 𝐺 = 𝐺 𝛿,ℓ has order 𝑛 = 𝛿ℓ 2 and √ 𝑖(𝐺) = 𝛿 + 𝛿(ℓ − 1) 2 = 𝛿ℓ 2 + 2𝛿 − 2𝛿ℓ = 𝑛 + 2𝛿 − 2 𝛿𝑛. As mention earlier, Favaron’s conjecture is true for 𝛿 = 0 and 𝛿 = 1. The conjecture was subsequently proven for 𝛿 = 2 in 1998 by Glebov and Kostochka [338]. The big breakthrough came in 1999 when Sun and Wang [699] proved the conjecture is true for all 𝛿. Before we present their proof, we shall need the following lemma. Consider a graph 𝐺 with minimum degree 𝛿. Let 𝑋 be an 𝛼-set of 𝐺 and let |𝑋 | = 𝑥. Let 𝑌 = 𝑉 \ 𝑋 and so, |𝑌 | = 𝑛 − 𝑥. Each vertex of 𝑋 has at least 𝛿 neighbors in 𝑌 and so, there are least 𝛿𝑥 edges between 𝑋 and 𝑌 in 𝐺. By the Pigeonhole Principle, there is a vertex in 𝑌 with at least 𝛿𝑥/(𝑛 − 𝑥) neighbors in 𝑋. For subsets 𝑋 and 𝑌 of vertices of a graph 𝐺, we denote the set of edges that join a vertex of 𝑋 and a vertex of 𝑌 in 𝐺 by 𝐺 [𝑋, 𝑌 ], or simply by [𝑋, 𝑌 ] if 𝐺 is clear from context. Thus, [𝑋, 𝑌 ] is the set of edges between 𝑋 and 𝑌 in 𝐺. Lemma 6.87 ([699]) Let 𝐺 be a graph with 𝛿(𝐺) = 𝛿. Let 𝑋 be an 𝛼-set of 𝐺 and let |𝑋 | = 𝑥. Let 𝑌 = 𝑉 \ 𝑋 and so, |𝑌 | = 𝑛 − 𝑥. Let 𝐵 be the bipartite graph with partite sets 𝑋 and 𝑌 , and whose edge set is 𝐺 [𝑋, 𝑌 ]. Let 𝑦 ′ be a vertex in 𝑌 such that deg 𝐵 (𝑦 ′ ) ≥ 𝛿𝑥/(𝑛 − 𝑥), and let 𝑌 ′ = 𝑦 ∈ 𝑌 : N 𝐵 (𝑦) ⊆ N 𝐵 (𝑦 ′ ) . If 𝑍 is a maximal independent set in 𝐺 [𝑌 ′ ] containing the vertex 𝑦 ′ , then the following hold: √ 𝛿𝑥 (a) If |𝑍 | − 𝛿 ≤ deg 𝐵 (𝑦 ′ ) − 𝑛−𝑥 , then 𝑖(𝐺) ≤ 𝑛 + 2𝛿 − 2 𝛿𝑛. 𝛿𝑥 ′ (b) If |𝑍 | − 𝛿 > deg 𝐵 (𝑦 ′ ) − 𝑛− 𝑥 , then for every proper subset 𝑍 of 𝑍, we have |𝑍 \ 𝑍 ′ | − 𝛿 > deg 𝐵 (𝑦 ′ ) − |N 𝐵 (𝑍 ′ )| − Proof (a) Assume that |𝑍 | − 𝛿 ≤ deg 𝐵 (𝑦 ′ ) − maximal independent set in 𝐺 and so,

𝛿𝑥 𝑛− 𝑥 .

𝛿𝑥 . 𝑛−𝑥


The set 𝑍 ∪ 𝑋 \ N 𝐵 (𝑦 ′ ) is a

Chapter 6. Upper Bounds in Terms of Minimum Degree

196


𝑖(𝐺) ≤ 𝑍 ∪ 𝑋 \ N 𝐵 (𝑦 ′ ) = |𝑍 | + |𝑋 | − |N 𝐵 (𝑦 ′ )| = 𝑥 + |𝑍 | − deg 𝐵 (𝑦 ′ ) 𝛿𝑥 ≤ 𝑥+𝛿− . 𝑛−𝑥 𝛿𝑥 By elementary calculus, the function 𝑓 (𝑥) = 𝑥 − 𝑛− 𝑥 is maximized when √ √
√ √ 𝑥 = 𝑛 − 𝛿𝑛. Further, 𝑓 𝑛 − 𝛿𝑛 = 𝑛 + 2𝛿 − 2 𝛿𝑛. Hence, 𝑖(𝐺) ≤ 𝑛 + 2𝛿 − 2 𝛿𝑛. 𝛿𝑥 ′ (b) Assume that |𝑍
| − 𝛿 > deg 𝐵 (𝑦 ′ ) − 𝑛− 𝑥 and let 𝑍 be a proper subset of 𝑍. The ′ ′ set 𝑍 ∪ 𝑋 \ N 𝐵 (𝑍 ) is a maximal independent set in 𝐺 and so, by the maximality of the set 𝑋,
|𝑍 ′ | + |𝑋 | − |N 𝐵 (𝑍 ′ )| = 𝑍 ′ ∪ 𝑋 \ N 𝐵 (𝑍 ′ ) ≤ |𝑋 |, implying that |𝑍 ′ | ≤ |N 𝐵 (𝑍 ′ )|.

Thus, by assumption,
𝛿𝑥 |𝑍 \ 𝑍 ′ | − 𝛿 = |𝑍 | − 𝛿 − |𝑍 ′ | > deg 𝐵 (𝑦 ′ ) − |N 𝐵 (𝑍 ′ )| − . 𝑛−𝑥 We now present a proof of Conjecture 6.86 due to Sun and Wang [699]. Theorem 6.88 ([699]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) = 𝛿, then √ 𝑖(𝐺) ≤ 𝑛 + 2𝛿 − 2 𝛿𝑛. Proof If 𝛿 = 0, then the bound simplifies to 𝑖(𝐺) √≤ 𝑛, which is obviously true. Suppose, to the contrary, that 𝑖(𝐺) > 𝑛 + 2𝛿 − 2 𝛿𝑛. Hence, we may assume that 𝛿 ≥ 1.√ An elementary calculus argument shows that the function
𝑓 (𝛿) = 𝑛 + 2𝛿 − 2 𝛿𝑛 is minimized when 𝛿 = 14 𝑛. Further, we note that 𝑓 14 𝑛 = 12 𝑛. √ Therefore, 𝑛 + 2𝛿 − 2 𝛿𝑛 ≥ 12 𝑛, with equality if and only if 𝛿 = 14 𝑛. If 𝛿 = 14 𝑛, then √ by Lemma 6.84, 𝑖(𝐺) ≤ 12 𝑛 = 𝑛 + 2𝛿 − 2 𝛿𝑛, a contradiction. Hence, 𝛿 ≠ 14 𝑛, which √ implies by our earlier observations that 𝑛 + 2𝛿 − 2 𝛿𝑛 > 12 𝑛. Let 𝑋 be an 𝛼-set of 𝐺 and let |𝑋 | = 𝑥. If 𝑥 ≤ 12 𝑛, then √ 𝑖(𝐺) ≤ 𝛼(𝐺) = 𝑥 ≤ 12 𝑛 < 𝑛 + 2𝛿 − 2 𝛿𝑛, a contradiction. Hence, 𝑥 > 12 𝑛. If 𝑌 = 𝑉 \ 𝑋, then |𝑌 | = 𝑛 − 𝑥. Let 𝐺 1 be the bipartite graph with partite sets ( 𝐴1 , 𝐵1 ), where 𝐴1 = 𝑋 and 𝐵1 = 𝑌 , and whose edge set consist of all edges of 𝐺 between 𝑋 and 𝑌 . The number of edges between 𝐴1 and 𝐵1 is at least 𝛿|𝑋 | = 𝛿𝑥 and so, by the Pigeonhole Principle, there is a vertex in 𝑦 1 ∈ 𝐵1 with at least 𝛿𝑥/|𝐵1 | = 𝛿𝑥/(𝑛 − 𝑥) neighbors in 𝑋. Thus, deg𝐺1 (𝑦 1 ) ≥ 𝛿𝑥/(𝑛 − 𝑥). Let  𝑌1 = 𝑦 ∈ 𝑌 : N𝐺1 (𝑦) ⊆ N𝐺1 (𝑦 1 ) , and let 𝑍1 be a maximal independent set in 𝐺 [𝑌1 ] containing the vertex 𝑦 1 . We 𝛿𝑥 note that 𝑍1 ⊆ 𝑌1 . If |𝑍1 | − 𝛿 ≤ deg𝐺1 (𝑦 1 ) − 𝑛− 𝑥 , then by Lemma 6.87(a), we have √ 𝑖(𝐺) ≤ 𝑛 + 2𝛿 − 2 𝛿𝑛, a contradiction. Hence,

Section 6.4. Bounds on the Independent Domination Number |𝑍1 | − 𝛿 > deg𝐺1 (𝑦 1 ) −

197

𝛿𝑥 . 𝑛−𝑥

(6.26)

Inequality (6.26) implies that |𝑍1 | − 𝛿 > 0. Let 𝐺 2 be the bipartite subgraph of 𝐺 1 induced by the set 𝐴2 ∪ 𝐵2 , where 𝐴2 = 𝑋 \ N𝐺1 (𝑦 1 ) and 𝐵2 = 𝑌 \ 𝑌1 . Thus, 𝐺 2 has partite sets 𝐴2 and 𝐵2 . We note that no vertex of 𝑌1 has a neighbor in 𝐺 that belongs to the set 𝐴2 . Thus, every neighbor in 𝐺 of a vertex in 𝐴2 belongs to the set 𝐵2 . Hence, for every vertex 𝑣 ∈ 𝐴2 , we have deg𝐺2 (𝑣) = deg𝐺 (𝑣) ≥ 𝛿. Claim 6.88.1 There exists a vertex 𝑦 2 ∈ 𝐵2 such that deg𝐺2 (𝑦 2 ) ≥

𝛿𝑥 𝑛− 𝑥 .

𝛿𝑥 Proof Suppose that deg𝐺2 (𝑦) < 𝑛− 𝑥 for all 𝑦 ∈ 𝐴2 . Counting the edges between 𝐴2 and 𝐵2 in 𝐺 2 and noting that |𝐵2 | = |𝑌 | − |𝑌1 | ≤ |𝑌 | − |𝑍1 | = 𝑛 − 𝑥 − |𝑍1 |, we have   ∑︁
𝛿𝑥 𝑛 − 𝑥 − |𝑍1 | > deg𝐺2 (𝑦) 𝑛−𝑥 𝑦 ∈ 𝐵2 ∑︁ = deg𝐺2 (𝑥) 𝑥 ∈ 𝐴2

≥ 𝛿| 𝐴2 |
= 𝛿 𝑥 − deg𝐺1 (𝑦 1 ) , or equivalently,

𝑥 𝑛 − 𝑥 − |𝑍1 | > 𝑥 − deg𝐺1 (𝑦 1 ) (𝑛 − 𝑥), which simplifies to 𝑥|𝑍1 | < (𝑛 − 𝑥) deg𝐺1 (𝑦 1 ).

(6.27)

By Inequalities (6.26) and (6.27), 

 𝛿𝑥 𝑥|𝑍1 | < (𝑛 − 𝑥) |𝑍1 | − 𝛿 + , 𝑛−𝑥 or equivalently,

𝑥 |𝑍1 | − 𝛿 < (𝑛 − 𝑥) |𝑍1 | − 𝛿 .

(6.28)

Since |𝑍1 | − 𝛿 > 0, Inequality (6.28) simplifies to 𝑥 < 𝑛 − 𝑥 and so, 𝑥 < contradicting our earlier assumption.

1 2 𝑛,

By Claim 6.88.1, we may assume there exists a vertex 𝑦 2 ∈ 𝐵2 such that deg𝐺2 (𝑦 2 ) ≥

𝛿𝑥 . 𝑛−𝑥

We now define the sets  𝑌 2 = 𝑦 ∈ 𝑌 : N𝐺1 (𝑦) ⊆ N𝐺1 (𝑦 2 ) , and  𝑌2 = 𝑦 ∈ 𝐵2 : N𝐺2 (𝑦) ⊆ N𝐺2 (𝑦 2 ) .

(6.29)

Chapter 6. Upper Bounds in Terms of Minimum Degree

198

We note that if 𝑦 ∈ 𝑌 2 , then either 𝑦 ∈ 𝑌1 or 𝑦 ∈ 𝑌2 and so, 𝑌 2 ⊆ 𝑌1 ∪ 𝑌2 . Let 𝑍 2 be a maximal independent set in 𝐺 [𝑌 2 ] containing the vertex 𝑦 2 , and let 𝑍2 be a maximal independent set in 𝐺 [𝑌2 ] containing the set 𝑍 2 ∩ 𝑌2 . We note that 𝑍 2 ⊆ 𝑌1 ∪ 𝑍2 and that the sets 𝑍1 and 𝑍2 are disjoint. By definition, 𝑦 2 ∈ 𝑍√2 . If 𝛿𝑥 |𝑍 2 | − 𝛿 ≤ deg𝐺1 (𝑦 2 ) − 𝑛− 𝑥 , then by Lemma 6.87(a), we have 𝑖(𝐺) ≤ 𝑛 + 2𝛿 − 2 𝛿𝑛, a contradiction. Hence, 𝛿𝑥 |𝑍 2 | − 𝛿 > deg𝐺1 (𝑦 2 ) − . (6.30) 𝑛−𝑥 Claim 6.88.2

|𝑍2 | − 𝛿 > deg𝐺2 (𝑦 2 ) −

𝛿𝑥 𝑛− 𝑥 .

Proof Since 𝐺 2 is an induced subgraph of 𝐺 1 , we note that deg𝐺1 (𝑦 2 ) ≥ deg𝐺2 (𝑦 2 ) and so, by Inequality (6.29), 𝛿𝑥 deg𝐺1 (𝑦 2 ) ≥ . 𝑛−𝑥 Let 𝑍 ′ = 𝑍 2 \ 𝑍2 . We note that 𝑍 2 \ 𝑍 ′ ⊆ 𝑍2 and so, |𝑍 2 \ 𝑍 ′ | ≤ |𝑍2 |. Since 𝑦 2 ∈ 𝑍 2 ∩ 𝑍2 , the set 𝑍 ′ is a proper subset of 𝑍 2 . Hence, by Inequality (6.30) and Lemma 6.87(b), 𝛿𝑥 |𝑍2 | − 𝛿 ≥ |𝑍 2 \ 𝑍 ′ | − 𝛿 > deg𝐺1 (𝑦 2 ) − |N𝐺1 (𝑍 ′ )| − . (6.31) 𝑛−𝑥 Since 𝑍 ′ ⊆ 𝑌1 , we note that N𝐺1 (𝑍 ′ ) ⊆ N𝐺1 (𝑦 2 ) \ N𝐺2 (𝑦 2 ) and so, |N𝐺1 (𝑍 ′ )| ≤ deg𝐺1 (𝑦 2 ) − deg𝐺2 (𝑦 2 ), or equivalently, deg𝐺1 (𝑦 2 ) − |N𝐺1 (𝑍 ′ )| ≥ deg𝐺2 (𝑦 2 ). Therefore, by Inequality (6.31), 𝛿𝑥 |𝑍2 | − 𝛿 > deg𝐺2 (𝑦 2 ) − . 𝑛−𝑥 Claim 6.88.2 implies that |𝑍2 | > 𝛿. More generally, for 𝑗 ≥ 2, we define 𝐺 𝑗 recursively to be the bipartite subgraph of 𝐺 𝑗 −1 induced by the set 𝐴 𝑗 ∪ 𝐵 𝑗 , where 𝐴𝑗 = 𝑋 \

𝑗 −1 Ø

N𝐺𝑖 (𝑦 𝑖 )

and

𝐵𝑗 = 𝑌 \

𝑖=1

𝑗 −1 Ø

𝑌𝑖 ,

𝑖=1

and 𝑦 𝑗 ∈ 𝐵 𝑗 such that deg𝐺 𝑗 (𝑦 𝑗 ) ≥

𝛿𝑥 . 𝑛−𝑥

Moreover,  𝑌 𝑗 = 𝑦 ∈ 𝑌 : N𝐺1 (𝑦) ⊆ N𝐺1 (𝑦 𝑗 ) ,  𝑌 𝑗 = 𝑦 ∈ 𝐵 𝑗 : N𝐺 𝑗 (𝑦) ⊆ N𝐺 𝑗 (𝑦 𝑗 ) , the set 𝑍 𝑗 is a maximal independent set in 𝐺 [𝑌 𝑗 ] containing vertex 𝑦 𝑗 , and the set 𝑍 𝑗 a maximal independent set in 𝐺 [𝑌 𝑗 ] containing the set 𝑍 𝑗 ∩ 𝑌 𝑗 such that |𝑍 𝑗 | − 𝛿 > deg𝐺 𝑗 (𝑦 𝑗 ) −

𝛿𝑥 . 𝑛−𝑥

(6.32)

Section 6.4. Bounds on the Independent Domination Number

199

We note that 𝑍𝑗 ⊆

Ø 𝑗 −1  𝑌𝑖 ∪ 𝑍 𝑗 , 𝑖=1

and that the sets 𝑍1 , 𝑍2 , . . . , 𝑍 𝑗 are pairwise disjoint and |𝑍𝑖 | > 𝛿 for all 𝑖 ∈ [ 𝑗]. Therefore, there exists an integer 𝑡 such that 𝑋=

𝑡 Ø

N𝐺𝑖 (𝑦 𝑖 ).

𝑖=1

We note that 𝑍1 ∪ 𝑍2 ∪ · · · ∪ 𝑍𝑡 ⊆ 𝑌 . Thus, since the sets 𝑍1 , 𝑍2 , . . . , 𝑍𝑡 are pairwise disjoint and |𝑍𝑖 | ≥ 𝛿 for all 𝑖 ∈ [𝑡], 𝑛 − 𝑥 = |𝑌 | ≥

𝑡 ∑︁

|𝑍𝑖 | ≥ 𝑡𝛿

𝑖=1

and so, 𝑡≤

𝑛−𝑥 . 𝛿

(6.33)

Since 𝑡 ∑︁

deg𝐺𝑖 (𝑦 𝑖 ) =

𝑖=1

𝑡 ∑︁

|N𝐺𝑖 (𝑦 𝑖 )| = |𝑋 | = 𝑥,

𝑖=1

by Inequality (6.32), |𝑌 | ≥

𝑡 ∑︁

|𝑍𝑖 | ≥

𝑖=1

∑︁ 𝑡

     𝛿𝑥 𝛿𝑥 deg𝐺𝑖 (𝑦 𝑖 ) + 𝑡 𝛿 − =𝑥+𝑡 𝛿− . 𝑛−𝑥 𝑛−𝑥 𝑖=1

(6.34)

By our earlier assumptions, we have 𝑛 < 2𝑥 and so, 𝛿−

𝛿𝑥 𝛿(𝑛 − 2𝑥) = < 0. 𝑛−𝑥 𝑛−𝑥

Thus, by Inequalities (6.33) and (6.34), 

   𝛿𝑥 𝑛−𝑥 𝛿𝑥 𝑛 = |𝑋 | + |𝑌 | > 2𝑥 + 𝑡 𝛿 − ≥ 2𝑥 + 𝛿− = 𝑛, 𝑛−𝑥 𝛿 𝑛−𝑥 a contradiction. As observed earlier, for every positive integer 𝛿, the bound in Theorem 6.88 is attained for infinitely many graphs. Hence, by Theorem 6.88 and the tightness of this bound, the independent domination number behaves different than the domination number.

200

Chapter 6. Upper Bounds in Terms of Minimum Degree

6.4.3 Regular Graphs As shown in the previous section, for any fixed minimum degree 𝛿 ≥ 1, there are graphs with 𝑖(𝐺) = 𝑛 − O (𝑛). However, the infinite class of graphs constructed earlier by Favaron [274] that achieve equality in the upper bound on the independent domination number given in Theorem 6.88 are far from regular, and the difference between their maximum and minimum degrees is large. If we require the graph to be regular, then the upper bound in Theorem 6.88 can be significantly improved. As first observed in 1964 by Rosenfeld [659], the independence number of a regular graph is at most one-half its order. Theorem 6.89 ([659]) For every integer 𝑟 ≥ 1, if 𝐺 is an 𝑟-regular connected graph of order 𝑛, then 𝛼(𝐺) ≤ 21 𝑛. Proof For 𝑟 ≥ 1, let 𝐺 be an 𝑟-regular connected graph of order 𝑛 and let 𝑋 be an 𝛼set of 𝐺. Let 𝑋 denote the complement of 𝑋 and so, 𝑋 = 𝑉 \ 𝑋. By double counting the
edges joining 𝑋 and its complement 𝑋, we have 𝑟 |𝑋 | = | [𝑋, 𝑋] | ≤ 𝑟 |𝑋 | = 𝑟 𝑛 − |𝑋 | and so, 𝑖(𝐺) ≤ 𝛼(𝐺) = |𝑋 | ≤ 12 𝑛. We note that equality in Theorem 6.89 is only obtainable for graphs with every component a balanced complete bipartite graph, as observed in [358]. Theorem 6.90 ([358]) For every integer 𝑟 ≥ 1, if 𝐺 is an 𝑟-regular connected graph of order 𝑛, then 𝑖(𝐺) ≤ 12 𝑛, with equality if and only if 𝐺 = 𝐾𝑟 ,𝑟 . Proof For 𝑟 ≥ 1, let 𝐺 be an 𝑟-regular connected graph of order 𝑛. Let 𝑋 be an 𝛼-set of 𝐺 and let 𝑋 =
𝑉 \ 𝑋. As shown in the proof of Theorem 6.89, we have 𝑟 |𝑋 | ≤ 𝑟 |𝑋 | = 𝑟 𝑛 − |𝑋 | and therefore 𝑖(𝐺) ≤ |𝑋 | ≤ 12 𝑛. Suppose that 𝑖(𝐺) = 12 𝑛. Hence, we must have equality throughout these two inequality chains, implying that |𝑋 | = |𝑋 | = 12 𝑛 and that each vertex in the complement 𝑋 of 𝑋 has exactly 𝑟 neighbors in 𝑋. Let 𝑋 = 𝑌 and so, 𝐺 is an 𝑟-regular, bipartite graph with partite sets 𝑋 and 𝑌 . Suppose, to the contrary, that 𝐺 ≠ 𝐾𝑟 ,𝑟 . Thus, there exist vertices 𝑥 ∈ 𝑋 and 𝑦 ∈ 𝑌 that are not adjacent in 𝐺. Let 𝑋1 be the set of all vertices in 𝑋 whose neighborhood is N𝐺 (𝑥), and let 𝑌1 be the set of all vertices in 𝑌 whose neighborhood is N𝐺 (𝑦), that is,  𝑋1 = 𝑣 ∈ 𝑋 : N𝐺 (𝑣) = N𝐺 (𝑥) , and  𝑌1 = 𝑣 ∈ 𝑌 : N𝐺 (𝑣) = N𝐺 (𝑦) . We note that 𝑥 ∈ 𝑋1 , 𝑦 ∈ 𝑌1 , and the set 𝑋1 ∪ 𝑌1 is an independent set in 𝐺. Let 𝑋2 = N𝐺 (𝑦) and let 𝑌2 = N𝐺 (𝑥) and so, |𝑋2 | = |𝑌2 | = 𝑟. By the regularity of 𝐺, we have |𝑋1 | ≤ 𝑟 and |𝑌1 | ≤ 𝑟. If |𝑋1 | = 𝑟, then 𝐺 [𝑋1 ∪ 𝑌2 ] = 𝐾𝑟 ,𝑟 , implying that the graph 𝐺 is disconnected, a contradiction. Hence, |𝑋1 | ≤ 𝑟 − 1. Analogously, |𝑌1 | ≤ 𝑟 − 1. Let 𝑆 = 𝑋1 ∪ 𝑌1 and so, |𝑆| = |𝑋1 | + |𝑌1 | ≤ 2(𝑟 − 1). Suppose that 𝑆 is a dominating set of 𝐺. In this case, 𝑆 is an ID-set of 𝐺. Further, 𝑋 = 𝑋1 ∪ 𝑋2 and 𝑌 = 𝑌1 ∪ 𝑌2 and so, 𝑛 = |𝑆| + 2𝑟 ≤ 4𝑟 − 2. Thus, 𝑖(𝐺) ≤ |𝑆| = 𝑛 − 2𝑟 ≤ 12 𝑛 − 1, a contradiction. Hence, 𝑆 is not a dominating set of 𝐺.

Section 6.4. Bounds on the Independent Domination Number

201

Let 𝑋3 = 𝑋 \ (𝑋1 ∪ 𝑋2 ) and let 𝑌3 = 𝑌 \ (𝑌1 ∪ 𝑌2 ). Each vertex in 𝑋3 has no neighbor in 𝑌1 and at most 𝑟 − 1 neighbors in 𝑌2 , and therefore has at least one neighbor in 𝑌3 . Analogously, each vertex in 𝑌3 has at least one neighbor in 𝑋3 . Let 𝐺 ′ be the subgraph of 𝐺 induced by the set 𝑋3 ∪ 𝑌3 . By our earlier observations, 𝐺 ′ is a bipartite
isolate-free graph
with partite sets 𝑋3 and 𝑌3 , implying that 𝑖(𝐺 ′ ) ≤ 21 |𝑋3 | + |𝑌3 | = 12 𝑛 − |𝑆| − 2𝑟 . A minimum ID-set in 𝐺 ′ can be extended to an ID-set of 𝐺 by adding the set 𝑆 to it. Hence, 𝑖(𝐺) ≤ |𝑆| + 𝑖(𝐺 ′ ) ≤ |𝑆| + 12 (𝑛 − |𝑆| − 2𝑟) = 12 𝑛 + 12 |𝑆| − 𝑟 ≤ 12 𝑛 + (𝑟 − 1) − 𝑟 = 12 𝑛 − 1, a contradiction. Hence, 𝐺 = 𝐾𝑟 ,𝑟 . Conversely, if 𝐺 = 𝐾𝑟 ,𝑟 , then 𝑛 = 2𝑟 and 𝑖(𝐺) = 𝑟 = 12 𝑛. We now consider regular graphs of fixed regularity. For 𝑟 ≥ 2 and 𝑛 ≥ 𝑟 +1, let G𝑟𝑛 denote the family of all connected 𝑟-regular graphs of order 𝑛 different from 𝐾𝑟 ,𝑟 . 𝑛 Further, let 𝑐𝑟 denote the supremum of 𝑖 (𝐺) 𝑛 taken over all graphs 𝐺 ∈ G𝑟 , that is, 𝑖(𝐺) . 𝐺 ∈ G𝑟𝑛 𝑛

𝑐𝑟 = sup

As a consequence of Theorem 6.90, we have the following result. Corollary 6.91 For 𝑟 ≥ 2, we have 𝑐𝑟 ≤ 12 . If 𝐺 is a connected 2-regular graph of order 𝑛, then 𝐺 is a cycle 𝐶𝑛 and 𝑖(𝐺) = 13 𝑛 . Hence, if 𝐺 is different from 𝐾2,2 , that is, if 𝑛 = 3 or 𝑛 ≥ 5, then we 3 have 𝑖 (𝐺) 𝑛 ≤ 7 , with equality if and only if 𝑛 = 7. This yields the following result. Proposition 6.92 The supremum 𝑐 2 = 37 . We next consider 3-regular graphs different from 𝐾3,3 . In 1999 Lam et al. [552] presented the best current general upper bound on the independent domination number of a cubic graph. We omit their proof which uses an intricate strong induction argument. Theorem 6.93 ([552]) If 𝐺 ≠ 𝐾3,3 is a connected cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 25 𝑛. The graphs 𝐾3,3 and 𝐶5 □ 𝐾2 are shown in Figure 6.27(a) and (b), respectively. As a consequence of Theorem 6.93 and the fact that equality in Theorem 6.93 holds for the 5-prism 𝐶5 □ 𝐾2 , we have the following result. Corollary 6.94 The supremum 𝑐 3 = 25 .

Chapter 6. Upper Bounds in Terms of Minimum Degree

202

(a) 𝐾3,3

(b) 𝐶5 □ 𝐾2

Figure 6.27 The graphs 𝐾3,3 and 𝐶5 □ 𝐾2

In 2013 Goddard and Henning [352] conjectured that the 25 -upper bound on the independent domination number of a 3-regular graph given in Theorem 6.93 can be improved if we forbid the exceptional graphs 𝐾3,3 and 𝐶5 □ 𝐾2 . Conjecture 6.95 ([352]) If 𝐺 ∉ {𝐾3,3 , 𝐶5 □ 𝐾2 } is a connected cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 38 𝑛. 1 2 Goddard and Henning [352] constructed two infinite families Fcubic and Fcubic of connected cubic graphs achieving the three-eighths upper bound as follows. For 1 𝑘 ≥ 1, a graph in the family Fcubic is constructed by taking two copies of the cycle 𝐶4𝑘 with respective vertex sequences 𝑎 1 𝑏 1 𝑐 1 𝑑1 . . . 𝑎 𝑘 𝑏 𝑘 𝑐 𝑘 𝑑 𝑘 and 𝑤 1 𝑥1 𝑦 1 𝑧 1 . . . 𝑤 𝑘 𝑥 𝑘 𝑦 𝑘 𝑧 𝑘 , and joining 𝑎 𝑖 to 𝑤 𝑖 , 𝑏 𝑖 to 𝑥𝑖 , 𝑐 𝑖 to 𝑧 𝑖 , and 𝑑𝑖 to 𝑦 𝑖 for each 𝑖 ∈ [𝑘]. For 2 ℓ ≥ 1, a graph in the family Fcubic is constructed by taking a copy of a cycle 𝐶3ℓ with vertex sequence 𝑎 1 𝑏 1 𝑐 1 . . . 𝑎 ℓ 𝑏 ℓ 𝑐 ℓ , and for each 𝑖 ∈ [ℓ], adding the vertices {𝑤 𝑖 , 𝑥𝑖 , 𝑦 𝑖 , 𝑧1𝑖 , 𝑧2𝑖 }, and joining 𝑎 𝑖 to 𝑤 𝑖 , 𝑏 𝑖 to 𝑥𝑖 , and 𝑐 𝑖 to 𝑦 𝑖 , and further for each 𝑗 1 𝑗 ∈ [2], joining 𝑧 𝑖 to each of the vertices 𝑤 𝑖 , 𝑥𝑖 , and 𝑦 𝑖 . Graphs in the families Fcubic 2 and Fcubic are illustrated in Figure 6.28(a) and (b), respectively.

(a) 𝐺

(b) 𝐻

1 2 and 𝐻 ∈ Fcubic Figure 6.28 Graphs 𝐺 ∈ Fcubic

1 2 Proposition 6.96 ([352]) If 𝐺 ∈ Fcubic ∪ Fcubic has order 𝑛, then 𝑖(𝐺) = 38 𝑛.

Section 6.4. Bounds on the Independent Domination Number

203

If Conjecture 6.95 is true, then the bound is tight as shown by Proposition 6.96. A graph operation that occurs frequently in the construction of extremal graphs is the expansion of a graph. If 𝑟 ≥ 1 an integer, the expansion exp(𝐺, 𝑟) of a graph 𝐺 is that graph obtained from 𝐺 by replacing each vertex 𝑣 of 𝐺 with an independent set 𝐼 𝑣 of cardinality 𝑟 and for every vertex 𝑣 in 𝐺, the open neighborhood of every vertex 𝑢 in 𝐼 𝑣 in the expansion of 𝐺 is given by Ø Nexp(𝐺,𝑟 ) (𝑢) = 𝐼𝑤 . 𝑤 ∈N(𝑣)

For example, for 𝑟 ≥ 1 the expansion exp(𝐶4 , 𝑟) of a 4-cycle is the complete bipartite graph 𝐾2𝑟 ,2𝑟 . We note that if 𝑥 and 𝑦 are two open twins in a graph 𝐺, that is, if N𝐺 (𝑥) = N𝐺 (𝑦), then any ID-set of 𝐺 contains either both 𝑥 and 𝑦 or neither of them. It follows that if 𝐷 is an ID-set in exp(𝐺, 𝑟), then for every vertex 𝑣 of 𝐺, the set 𝐷 either contains all of 𝐼 𝑣 or none of 𝐼 𝑣 . Furthermore, the set {𝑣 : 𝐼 𝑣 ⊆ 𝐷} is an ID-set of the graph 𝐺, implying that 𝑖(𝐺) ≤ 𝑟1 · 𝑖(exp(𝐺, 𝑟)), or equivalently, Ð 𝑖(exp(𝐺, 𝑟)) ≥ 𝑟 · 𝑖(𝐺). On the other hand, if 𝐼 is an ID-set of 𝐺, then the set 𝑣 ∈𝐷 𝐼 𝑣 is an ID-set in exp(𝐺, 𝑟), implying that 𝑖(exp(𝐺, 𝑟)) ≤ 𝑟 · 𝑖(𝐺). Consequently, 𝑖(exp(𝐺, 𝑟)) = 𝑟 · 𝑖(𝐺). We state this formally as follows. Lemma 6.97 If 𝐺 is a graph and 𝑟 ≥ 1 an integer, then 𝑖(exp(𝐺, 𝑟)) = 𝑟 · 𝑖(𝐺). To illustrate Lemma 6.97, consider the expansion exp(𝐶7 , 2) of a 7-cycle, illustrated in Figure 6.29. By Lemma 6.97, we have 𝑖(exp(𝐶7 , 2)) = 2 · 𝑖(𝐶7 ) = 2 · 3 = 6. Hence, the expansion 𝐺 = exp(𝐶7 , 2) of a 7-cycle is a connected 4-regular graph of order 𝑛 = 14 satisfying 𝑖(𝐺) = 6 = 37 𝑛, implying that the constant 𝑐 4 ≥ 37 .

Figure 6.29 The expansion exp(𝐶7 , 2) In 2013 Goddard and Henning [352] conjectured that if 𝐺 ≠ 𝐾4,4 is a connected 4-regular graph of order 𝑛, then 𝑖(𝐺) ≤ 37 𝑛. Equivalently, they conjectured that 𝑐 4 = 37 . In 2021 Cho et al. [171] announced they had settled this conjecture in the affirmative. Theorem 6.98 ([171]) The supremum 𝑐 4 = 37 . It would be interesting to determine the exact value of the constant 𝑐𝑟 for all 𝑟 ≥ 5. However, the question of best possible bounds of the independent domination number

Chapter 6. Upper Bounds in Terms of Minimum Degree

204

of connected 𝑟-regular graphs different from 𝐾𝑟 ,𝑟 for 𝑟 ≥ 5 remains unresolved, even for the special case when 𝑟 = 5. The following result was observed in [352]. Lemma 6.99 ([352]) For all positive integers 𝑟 and 𝑠, 𝑐𝑟 𝑠 ≥ 𝑐𝑟 . Proof Let 𝐺 be a connected graph in G𝑟𝑛 that gives the value for 𝑐𝑟 . The expansion 𝐻 = exp(𝐺, 𝑠) of the graph 𝐺 is a connected (𝑟 𝑠)-regular graph of order 𝑛(𝐻) = 𝑛 × 𝑠 satisfying 𝑖(𝐻) 𝑠 × 𝑖(𝐺) 𝑖(𝐺) = = , 𝑛(𝐻) 𝑛×𝑠 𝑛 implying that 𝑐𝑟 𝑠 ≥ 𝑐𝑟 . Goddard and Henning [352] posed the following question. Question 6.100 Is it true that the supremum 𝑐𝑟 tends to

1 2

as 𝑟 → ∞?

Question 6.100 was answered in the affirmative in 2020 by Blumenthal [82] in his PhD thesis. In order to construct connected graphs 𝐺 in the family G𝑟𝑛 such that 𝑖 (𝐺) 1 𝑘 𝑛 ≥ 2 − 𝜀 for every 𝜀 > 0, for 𝑟 > 𝑘 ≥ 2, let 𝐾𝑟 ,𝑟 be the graph obtained from a complete bipartite graph 𝐾𝑟 ,𝑟 by selecting an arbitrary vertex of the graph, which we call the gluing vertex, and removing 𝑘 − 1 edges incident with this vertex. The 4 with gluing vertex 𝑥 is illustrated in Figure 6.30. graph 𝐾5,5

𝑥 4 with gluing vertex 𝑥 Figure 6.30 The graph 𝐾5,5

For integers 𝑟 > 𝑘 ≥ 2 with 𝑘 even, let G𝑘,𝑟 be the family of graphs 𝐺 𝑘,𝑟 constructed as follows. Let 𝐺 𝑘,𝑟 be obtained from 𝑘 vertex-disjoint copies of 𝐾𝑟𝑘,𝑟 by adding all edges between the 𝑘 gluing vertices, so that the gluing vertices form a clique 𝐾 𝑘 . Let 𝑀 be an arbitrary perfect matching in this complete graph of (even) order 𝑘 consisting of the gluing vertices. For each gluing vertex 𝑣, let 𝐺 𝑣 be the copy of 𝐾𝑟𝑘,𝑟 that contains 𝑣, and let 𝑁 𝑣 be the set of vertices of degree 𝑟 − 1 in 𝐺 𝑣 different from 𝑣. We note that |𝑁 𝑣 | = 𝑘 − 1 and that the 𝑘 − 1 edges joining 𝑣 to vertices in 𝑁 𝑣 were deleted when constructing 𝐺 𝑣 . Further, we note that in 𝐺 𝑣 every vertex has degree 𝑟, except for the vertex 𝑣 which has degree 𝑟 − 𝑘 + 1 and the vertices in 𝑁 𝑣 which have degree 𝑟 − 1. For each edge 𝑢𝑣 ∈ 𝑀, we add a perfect matching between the vertices of 𝑁𝑢 and 𝑁 𝑣 . Let 𝐺 𝑘,𝑟 be the resulting 𝑟-regular graph of order 2𝑟 𝑘 and let G𝑘,𝑟 be the family of all such graphs 𝐺 𝑘,𝑟 . A graph in the family G4,5 is illustrated in Figure 6.31. Blumenthal [82] proved if 𝐺 ∈ G𝑘,𝑟 , then 𝑖(𝐺) ≥ 𝑘 + (𝑟 − 𝑘 − 1) (𝑘 − 1) = 𝑟 (𝑘 − 1) − 𝑘 2 + 𝑘 + 1. With a more detailed analysis, we can determine precisely the independent domination number of a graph in the family G𝑘,𝑟 .

Section 6.4. Bounds on the Independent Domination Number

205

Figure 6.31 A graph in the family G4,5

Proposition 6.101 For 𝑟 > 𝑘 ≥ 2 with 𝑘 even, if 𝐺 ∈ G𝑘,𝑟 , then 𝑖(𝐺) = 𝑟 (𝑘 − 1) − 1 2 𝑘 (𝑘 − 3). Proof Let 𝐺 ∈ G𝑘,𝑟 and let 𝐴 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } be the set of 𝑘 gluing vertices used in the construction of 𝐺. Let 𝐺 𝑖 be the copy of 𝐾𝑟𝑘,𝑟 that contains the gluing vertex 𝑣 𝑖 for 𝑖 ∈ [𝑘]. Renaming vertices if necessary, we may assume that the perfect matching 𝑀 in the complete graph 𝐺 [ 𝐴] = 𝐾 𝑘 used in the construction of 𝐺 is 𝑀=

𝑘/2 Ø

{𝑣 2𝑖−1 , 𝑣 2𝑖 }.

𝑖=1

Let   𝐺 2𝑖−1,2𝑖 be the subgraph of 𝐺 induced by the set 𝑉 (𝐺 2𝑖−1 ) ∪ 𝑉 (𝐺 2𝑖 ) for 𝑖 ∈ 𝑘2 . We note that the subgraphs 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑘/2 are vertex-disjoint. Further, the only edges of 𝐺 joining a vertex in one such subgraph to a vertex in another such subgraph are edges joining two link vertices. Let 𝐼 be an 𝑖-set of 𝐺. We note that 𝐺 [ 𝐴] is a complete graph 𝐾 𝑘 and so, the independent set 𝐼 contains at most one vertex from the set 𝐴, that   is, the set 𝐼 contains at most one gluing vertex. Let 𝐼𝑖 = 𝐼 ∩ 𝑉 (𝐺 2𝑖−1,2𝑖 ) for 𝑖 ∈ 𝑘2 . If (exactly) one of 𝑣 2𝑖−1 and 𝑣 2𝑖 belongs to the set 𝐼, then |𝐼𝑖 | ≥ 𝑘 + (𝑟 − 𝑘 + 1) = 𝑟 + 1. If neither 𝑣 2𝑖−1 nor 𝑣 2𝑖 belongs to the set 𝐼, then |𝐼𝑖 | ≥ 𝑟 + (𝑟 − 𝑘 + 1) = 2𝑟 − 𝑘 + 1. Since the set 𝐼 contains at most one gluing vertex, this implies that 𝑖(𝐺) = |𝐼 | =

𝑘/2 ∑︁

|𝐼𝑖 | ≥ 𝑟 + 1 + (2𝑟 − 𝑘 + 1)

1 2𝑘


− 1 = 𝑟 (𝑘 − 1) − 12 𝑘 (𝑘 − 3). (6.35)

𝑖=1

We construct next an independent dominating set 𝐷 of 𝐺 satisfying |𝐷| ≤ 𝑟 (𝑘 − 1) − 12 𝑘 (𝑘 − 3). For 𝑗 ∈ [𝑘], let 𝑋 𝑗 be the set of neighbors of 𝑣 𝑗 in 𝐺 𝑗 and let 𝑌 𝑗 be the set of vertices at distance 3 from 𝑣 𝑗 in 𝐺 𝑗 . We note that 𝑌 𝑗 is the set of vertices in 𝐺 𝑗 of degree 𝑟 − 1 that are different from 𝑣 𝑗 , and that |𝑋 𝑗 | = 𝑟 − 𝑘 + 1 and |𝑌 𝑗 | = 𝑘 − 1. Let Ø  Ø  𝑘/2 𝑘 𝐷 = {𝑣 1 } ∪ 𝑌1 ∪ 𝑋𝑖 ∪ 𝑌2𝑖 . 𝑖=2

𝑖=2

Chapter 6. Upper Bounds in Terms of Minimum Degree

206

The set 𝐷 is an independent dominating set of 𝐺 satisfying 𝑖(𝐺) ≤ |𝐷 | = 1 + |𝑌1 | +

𝑘 ∑︁ 𝑖=2

|𝑋𝑖 | +

𝑘/2 ∑︁

|𝑌2𝑖 |

𝑖=2

= 1 + (𝑘 − 1) + (𝑘 − 1) (𝑟 − 𝑘 + 1) + (𝑘 − 1)

𝑘 2

−1




= 𝑟 (𝑘 − 1) − 12 𝑘 (𝑘 − 3). Consequently, by Inequality (6.35), we have 𝑖(𝐺) = 𝑟 (𝑘 − 1) − 12 𝑘 (𝑘 − 3). To illustrate Proposition 6.101, if 𝐺 ∈ G4,5 , then 𝑖(𝐺) = 13. An example of an 𝑖-set in the graph 𝐺 ∈ G4,5 shown in Figure 6.31 is given by the set of 13 highlighted vertices. We are now in a position to state the following result due to Blumenthal [82]. However, the proof we present follows from Proposition 6.101. Theorem 6.102 ([82]) The values of 𝑐𝑟 tend to

1 2

as 𝑟 → ∞.

Proof For integers 𝑟 and 𝑘, where 𝑟 > 𝑘 ≥ 2 with 𝑘 even, let 𝐺 ∈ G𝑘,𝑟 be a graph of order 𝑛. Let 𝑓𝑟 (𝑘) be the function defined by 𝑓𝑟 (𝑘) =

1 1 𝑘 3 − − + . 2 2𝑘 4𝑟 4𝑟

By Proposition 6.101 and by the definition of the constant 𝑐𝑟 , 𝑐𝑟 ≥

1 𝑖(𝐺) 𝑟 (𝑘 − 1) − 2 𝑘 (𝑘 − 3) = 𝑓𝑟 (𝑘). = 𝑛 2𝑟 𝑘

Thus, 𝑐𝑟 ≥ 𝑓𝑟 (𝑘) for all integer values of 𝑘, where 2 ≤ 𝑘 ≤ 𝑟 − 1 and 𝑘 is even. For real optimization with 𝑟 ≥ 3 a fixed integer and 𝑘√a real number √ (where 2 ≤ 𝑘 ≤ 𝑟 − 1), the function 𝑓𝑟 (𝑘) is maximized when 𝑘 = 2𝑟. Thus, if 2𝑟 is an even integer, then this yields 𝑐 𝑟 ≥ 𝑓𝑟

√
1 3 1 2𝑟 = + −√ . 2 4𝑟 2𝑟

(6.36)

For integer optimization with 𝑟 ≥ 3 a fixed integer and 𝑘 an even integer satisfying 2 ≤ 𝑘 ≤ 𝑟 − 1, the maximum value of the √ function 𝑓𝑟 (𝑘) is max{𝑘 1 , 𝑘 2 }, where 𝑘 1 and 𝑘 2 are even integers such that 𝑘 1 ≤ 2𝑟 ≤ 𝑘 2 and 𝑘 2 = 𝑘 1 + 2. We remark that sometimes the maximum value of 𝑓𝑟 (𝑘) is attained at 𝑘 = 𝑘 1 and sometimes the maximum value of 𝑓𝑟 (𝑘) is attained at 𝑘 = 𝑘 2 . In any event, this yields a lower bound of 𝑐𝑟 of approximately 12 + 4𝑟3 − √1 , which tends to 12 as 𝑟 → ∞. By Corollary 6.91, 2𝑟 we have 𝑐𝑟 ≤ 12 . Consequently, 𝑐𝑟 tends to 12 as 𝑟 → ∞. We next consider the independent domination number in regular graphs of larger degree. Favaron [274] was the first to improve the upper bound of Theorem 6.89 for 𝛿 ≥ 12 𝑛.

Section 6.4. Bounds on the Independent Domination Number

207

Theorem 6.103 ([274]) If 𝐺 is a 𝛿-regular graph of order 𝑛 with 𝛿 ≥ 21 𝑛, then 𝑖(𝐺) ≤ 𝑛 − 𝛿, with equality only for complete multipartite graphs with all partite sets of the same order. Haviland [398–400] improved the upper bound √ of
Theorem 6.89 for values of 𝛿 in the range 14 𝑛 ≤ 𝛿 ≤ 12 𝑛. We remark that 12 3 − 5 ≈ 0.3820. Theorem 6.104 ([398–400]) If 𝐺 is a 𝛿-regular graph of order 𝑛 with 𝛿 ≤ then ( √
√ 𝑛 − 𝑛𝛿 if 14 𝑛 ≤ 𝛿 ≤ 12 3 − 5 𝑛 𝑖(𝐺) ≤ √
𝛿 if 12 3 − 5 𝑛 ≤ 𝛿 ≤ 12 𝑛.

1 2 𝑛,

√ We remark that in the statement of Theorem 6.104, the two values 𝑛 − 𝑛𝛿 and 𝛿 √
are the same when 𝛿 = 12 3 − 5 𝑛. This bound in Theorem 6.104 was subsequently improved for 𝛿 ≥ 25 𝑛 by Goddard et al. [358]. Theorem 6.105 ([358]) If 𝐺 is a 𝛿-regular graph of order 𝑛 with 25 𝑛 ≤ 𝛿 < 12 𝑛, then 𝑖(𝐺) ≤ 23 (𝑛 − 𝛿). For small regularity, namely 𝛿 < 14 𝑛, Haviland [402] established the following bounds. Theorem 6.106 ([402]) If 𝐺 is a 𝛿-regular graph of order 𝑛 with 𝛿 < 14 𝑛, then 1   𝑛    2 3 𝑖(𝐺) ≤ 5 (𝑛 − 𝛿)     2𝛿 

if 1 ≤ 𝛿 ≤ 16 𝑛 if if

1 3 6 𝑛 ≤ 𝛿 ≤ 13 𝑛 3 1 13 𝑛 ≤ 𝛿 ≤ 4 𝑛.

We remark that in the statement of Theorem 6.106, the two values 12 𝑛 and 35 (𝑛 − 𝛿) are the same when 𝛿 = 16 𝑛. Moreover, the two values 35 (𝑛 − 𝛿) and 2𝛿 are the same 3 when 𝛿 = 13 𝑛. 1 For 𝛿 < 4 𝑛, Haviland [402] established the following upper bounds on 𝑖(𝐺) for 𝛿-regular graphs. Theorem 6.107 ([402]) If 𝐺 is a 𝛿-regular graph of order 𝑛 with 14 𝑛 ≤ 𝛿 ≤ 25 𝑛, then √ (1 √

2 2 if 𝛿 < 12 33 − 5 𝑛 2 3𝑛 + 2𝛿 − 𝑛 + 4𝛿 + 20𝑛𝛿 𝑖(𝐺) ≤ √
𝛿 if 𝛿 ≥ 12 33 − 5 𝑛. We close this section with the following result of Lyle [576] who studied a structural approach for independent domination in graphs. If 𝐺 is a 𝛿-regular graph on 𝑛 vertices with 𝛿 < 12 𝑛, then its complement 𝐺 is always connected.

Chapter 6. Upper Bounds in Terms of Minimum Degree

208

Theorem 6.108 ([576]) For every integer 𝑘 ≥ 4, if 𝐺 is a 𝛿-regular graph of order 𝑛 such that 𝐺 is connected, then  3   5 (𝑛 − 𝛿)      1𝑛 𝑖(𝐺) ≤ 2 5    8 (𝑛 − 𝛿)    2𝑛 𝑘

6.5

if

1 6𝑛

< 𝛿 < 14 𝑛

if 𝛿 = 14 𝑛 1 4𝑛

< 𝛿 < 𝑛 − 8 and 𝛿 ≠
if 𝛿 = 𝑘−3 𝑘 𝑛. if

𝑘−3 𝑘




𝑛

Summary

In this chapter, we have presented upper bounds on the domination number, the total domination number, and the independent domination number, in terms of the order 𝑛 and minimum degree 𝛿 of the graph. For a connected graph of order 𝑛 ≥ 3, we compare these best known upper bounds for small minimum degree 𝛿 ∈ [6] in Table 6.13, where Bdom and Btdom are families of seven and six, respectively, exceptional graphs of small orders.

Bound on domination parameter 𝛿≥

𝛾(𝐺) ≤

𝛾t (𝐺) ≤

1

1 2𝑛

2 3𝑛

2 3 4 5 6

2 5𝑛

if 𝐺 ∉ Bdom 3 8𝑛 4 11 𝑛 1 3𝑛 127 418 𝑛

4 7𝑛

if 𝐺 ∉ Btdom

4 11 4 13

1 2𝑛 3 7𝑛
11 + 800 𝑛
17 + 494 𝑛

𝑖(𝐺) ≤ √ 𝑛+2−2 𝑛 √ 𝑛 + 4 − 2 2𝑛 √ 𝑛 + 6 − 2 3𝑛 √ 𝑛 + 8 − 2 4𝑛 √ 𝑛 + 10 − 2 5𝑛 √ 𝑛 + 12 − 2 6𝑛

Table 6.13 A summary of upper bounds for small 𝛿 ∈ [6]

Chapter 7

Probabilistic Bounds and Domination in Random Graphs 7.1 Introduction In Chapter 6, we presented upper bounds on the domination number of a graph in terms of its order 𝑛 and minimum degree 𝛿. For small 𝛿, the best known bounds to date are summarized in Table 6.5 in Chapter 6. Recall that for 𝛿 ∈ [3], the bounds given in the table are tight, while for all values of 𝛿 ≥ 4, no tight bound on the domination number is yet known. When 𝛿 is sufficiently large, optimal bounds on the domination number can be found using the Probabilistic Method. We present several such probabilistic bounds, including one for the total domination number, in this chapter. We also show that if we carefully choose the probability 𝑝 that an edge is chosen in a random graph of order 𝑛, then the domination numbers √︁ and total domination √︁ enjoy a tight concentration, roughly between √1 𝑛 ln(𝑛) and √1 𝑛 ln(𝑛). We give 2 2 2 similar results for the independent domination number.

7.2

Probabilistic Bounds

One of the earliest bounds on the domination number was due to Arnautov [36] in 1974 and Payan [632] in 1975. Theorem 7.1 ([36, 632]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then 𝛿+1 𝑛 ∑︁ 1 𝛾(𝐺) ≤ . 𝛿 + 1 𝑗=1 𝑗 The proof of Theorem 7.1 presented in [36, 632] appears to follow a careful © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_7

209

210

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

analysis of a greedy algorithm to find a dominating set. Since the 𝑘 th harmonic number, 𝑘 ∑︁ 1 𝐻𝑘 = , 𝑗 𝑗=1 1 is approximately Φ + ln(𝑘) + 2𝑘 , where Φ = 0.57721 . . . is the Euler-Mascheroni constant, as an immediate consequence of Theorem 7.1 we have the following upper bound on the domination number of a graph in terms of its order and minimum degree.

Theorem 7.2 ([36, 632]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then   1 + ln(𝛿 + 1) 𝛾(𝐺) ≤ 𝑛. 𝛿+1 We present here a proof of the upper bound in Theorem 7.2 using the probabilistic method. The idea of the proof is to generate a random set of vertices in a given graph where each vertex is chosen with a certain probability. Since the random choice of our set is not necessarily a dominating set, the set of vertices not yet dominated is added to this random set to form a dominating set. By carefully choosing the probability that each vertex is included in the initial random set, we obtain an upper bound on the domination number. Probabilistic Proof of Theorem 7.2 Let 𝑅 be a random subset of vertices of a graph 𝐺 with minimum degree 𝛿 ≥ 1, where a vertex is chosen to be in 𝑅 with probability 𝑝 and independently of the choice for any other vertex, where 𝑝=

ln(𝛿 + 1) . 𝛿+1

Let 𝑆 be the set of vertices in 𝐺 outside 𝑅 that have no neighbor in 𝑅, that is,  𝑆 = 𝑣 ∈ 𝑉 \ 𝑅 : N(𝑣) ∩ 𝑅 = ∅ . The set 𝑅 ∪ 𝑆 is a dominating set of 𝐺. The expected value of |𝑅| is E(|𝑅|) = 𝑛𝑝. The random variable |𝑆| can be written as the sum of 𝑛 indicator random variables 𝑋𝑣 (𝑆) for each 𝑣 ∈ 𝑉, where 𝑋𝑣 (𝑆) = 1 if 𝑣 ∈ 𝑆 and 𝑋𝑣 (𝑆) = 0 otherwise. For each vertex 𝑣 ∈ 𝑉, the expected value of 𝑋𝑣 (𝑆) is the probability that 𝑣 and its neighbors are not in 𝑅; that is,
E 𝑋𝑣 (𝑆) = (1 − 𝑝) deg(𝑣)+1 ≤ (1 − 𝑝) 𝛿+1 since deg(𝑣) ≥ 𝛿 and 0 ≤ 1 − 𝑝 ≤ 1. Using the inequality 1 − 𝑥 ≤ 𝑒 −𝑥 for 𝑥 a real number,
E 𝑋𝑣 (𝑆) ≤ (1 − 𝑝) 𝛿+1 ≤ 𝑒 − 𝑝 ( 𝛿+1)

Section 7.2. Probabilistic Bounds

211

for each vertex 𝑣 in 𝐺. Thus, by linearity of expectation,


E |𝑅 ∪ 𝑆| = E |𝑅| + E |𝑆| ∑︁
E 𝑋𝑣 (𝑆) ≤ 𝑛𝑝 + 𝑣 ∈𝑉

≤ 𝑛𝑝 + 𝑛𝑒 − 𝑝 ( 𝛿+1)   1 + ln(𝛿 + 1) = 𝑛. 𝛿+1 Since expectation is an average value, there is a set 𝑅 and an associated set 𝑆 such that 𝑅 ∪ 𝑆 is a dominating set in 𝐺 and   1 + ln(𝛿 + 1) |𝑅| + |𝑆| ≤ 𝑛. 𝛿+1 Since 𝛾(𝐺) ≤ |𝑅| + |𝑆|, this completes the proof of Theorem 7.2. The bound on the domination number in Theorem 7.2 is not very good for small values of 𝛿. However, as 𝛿 increases, the bound gets increasingly tight. Indeed, in 1990 Alon [17] provided a probabilistic proof that shows that the bound in Theorem 7.2 is asymptotically optimal, that is, when 𝛿 → ∞. Many probabilistic bounds on the domination number of a graph have been established over the past few decades. Using probabilistic arguments, in 1985 Caro and Roditty [134] (see also [135]) established the following upper bound on the domination number of a graph that is valid for all 𝛿 ≥ 1. Theorem 7.3 ([134]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then 

1 𝛾(𝐺) ≤ 1 − 𝛿 𝛿+1

 1+ 𝛿1 ! 𝑛.

In 1998 Clark et al. [180] established the following probabilistic upper bound on the domination number. Theorem 7.4 ([180]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then  𝛿+1 Ö 𝛾(𝐺) ≤ 1 − 𝑗=1

 𝑗𝛿 𝑛. 𝑗𝛿 + 1

The bound given in Theorem 7.4 is better than the bound given in Theorem 7.2 for all 𝛿 ≥ 5. Recall that the results presented in Chapter 6 showed that Conjecture 6.28 holds, that is, if 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then   𝛿 𝛾(𝐺) ≤ 𝑛, 3𝛿 − 1

212

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

unless 𝛿 = 2 and 𝐺 is one of the seven graphs in the family Bdom shown in Figure 6.1. We note that the bound given in Theorem 7.4 is better than the bound in Conjecture 6.28 for all 𝛿 ≥ 7. In 2012 Biró et al. [78] further improved the upper bound on the domination number and proved the following result. Theorem 7.5 ([78]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then ! 𝛿2 − 𝛿 + 1 𝛾(𝐺) ≤ 1 −
𝑛. Î 𝛿+1 1 + 𝛿 𝛿−1 𝑗=1 1 + 𝑗 𝛿 The bound given in Theorem 7.5 is better than the bound given in Theorem 7.4 for all 𝛿 ≥ 1 and is better than the bound given in Theorem 7.2 for all 𝛿 ≥ 6. In 1999 Harant et al. [380] used probabilistic arguments to obtain upper bounds on the domination number of a graph. Theorem 7.6 ([380]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then
  𝛿1 ! ∑︁  1  𝛿1 deg(𝑣)+1 1 𝛾(𝐺) ≤ 1 − 𝛿 . 𝑛+ 𝛿 𝛿+1 𝑣 ∈𝑉 Further, a dominating set of cardinality at most the expression on the right hand side
of the above inequality can be constructed in O Δ2 𝑛 time. Theorem 7.7 ([380]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then   𝛿1   𝛿1 Ö   𝛿1 ! ∑︁ 1 1 1 𝛾(𝐺) ≤ 1− + . deg(𝑣) + 1 deg(𝑣) + 1 deg(𝑢) + 1 𝑣 ∈𝑉 𝑢∈N(𝑣)

In the special case when 𝐺 is a regular graph, the bounds in Theorems 7.6 and 7.7 simplify as follows. Theorem 7.8 ([380]) If 𝐺 is a 𝛿-regular graph of order 𝑛, then ! 𝛿 𝛾(𝐺) ≤ 1 − 𝑛. 1 (𝛿 + 1) 1+ 𝛿 In 2019 Jafari Rad [509] gave a new probabilistic upper bound on the domination number that improves the previous bounds. Before stating and proving the result due to Jafari Rad, we present two key preliminary lemmas. Lemma 7.9 ([509]) Let 𝐺 be a graph of order 𝑛 with minimum degree 𝛿 > 1 and maximum degree Δ, and let 0 < 𝑝 < 1. Let 𝐴 be a subset of vertices of 𝐺, where a vertex is chosen to be in 𝐴 with probability 𝑝 and independently of the choice for any other vertex. Let 𝐴′′ ⊆ 𝐴′ ⊆ 𝐴 be defined by   𝐴′ = 𝑣 ∈ 𝑉 : N[𝑣] ⊆ 𝐴 and 𝐴′′ = 𝑣 ∈ 𝑉 : N[𝑣] ⊆ 𝐴′ .

Section 7.2. Probabilistic Bounds

213

For any integer 𝑠 ≥ 1, there is a subset 𝑆 ⊆ 𝐴′ such that 𝑆 dominates 𝐴′′ and |𝑆| ≤ 𝑓 (𝑠 − 1)| 𝐴′ |, where 𝑓 (0) = 𝑝 + (1 − 𝑝) 1+ 𝛿 , and 𝑓 (0)

1− 𝑓 (0)

{ ∑︁ }| }| z { z 𝑗
𝑓 ( 𝑗) = 𝑝 + (1 − 𝑝) 1+ 𝛿 − 1 − 𝑝 − (1 − 𝑝) 1+ 𝛿 𝑝 𝑖 (Δ+1) 𝑖=1

for every integer 𝑗 ≥ 1. Proof We proceed by induction on 𝑠 ≥ 1. To prove the base case, we show that there is a subset 𝑆 ⊆ 𝐴 such that 𝑆 dominates 𝐴′′ and
|𝑆| ≤ 𝑓 (0)| 𝐴′ | = 𝑝 + (1 − 𝑝) 1+ 𝛿 | 𝐴′ |. Let 𝐴1 be a subset of vertices of 𝐴′ , where a vertex is chosen to be in 𝐴1 with probability 𝑝 and independently of the choice for any other vertex. Let 𝐵1 ⊆ 𝐴′′ be the set of vertices in 𝐴′′ that are not dominated by 𝐴1 . We now let 𝑆1 = 𝐴1 ∪ 𝐵1 and note that the set 𝑆1 dominates 𝐴′′ . The expected value of | 𝐴1 | is
E | 𝐴1 | = | 𝐴′ | 𝑝. By definition, we note that if 𝑣 ∈ 𝐴′′ , then 𝑣 and its neighbors belong to the set 𝐴′ , and so deg𝐺 (𝑣) = deg𝐺 [ 𝐴′ ] (𝑣). Thus, Pr(𝑣 ∈ 𝐵1 ) = (1 − 𝑝) 1+deg𝐺 [ 𝐴′ ] (𝑣) = (1 − 𝑝) 1+deg𝐺 (𝑣) ≤ (1 − 𝑝) 1+ 𝛿 , implying that


E |𝐵1 | ≤ | 𝐴′ |(1 − 𝑝) 1+ 𝛿 .

Thus, by linearity of expectation,



E |𝑆1 | = E | 𝐴1 ∪ 𝐵1 | = E | 𝐴1 | + E |𝐵1 | ≤ | 𝐴′ | 𝑝 + | 𝐴′ | (1 − 𝑝) 1+ 𝛿
= 𝑝 + (1 − 𝑝) 1+ 𝛿 | 𝐴′ | = 𝑓 (0)| 𝐴′ |. Since expectation is an average value, there is a subset 𝑆 ⊆ 𝐴′ such that 𝑆 dominates 𝐴′′ and |𝑆| ≤ 𝑓 (0)| 𝐴′ |. This establishes the base case. For the inductive hypothesis, let 𝑠 ≥ 1 and assume the result holds for all positive integers 𝑠′ , where 𝑠′ ≤ 𝑠. We show that the result holds for 𝑠 + 1. As before, let 𝐴1 be a subset of vertices of 𝐴′ , where a vertex is chosen to be in 𝐴1 with probability 𝑝 and independently of the choice for any other vertex, and let 𝐵1 ⊆ 𝐴′′ be the set of vertices in 𝐴′′ that are not dominated by 𝐴1 . Let 𝐴1′′ ⊆ 𝐴1′ ⊆ 𝐴1 be defined by  𝐴1′ = 𝑣 ∈ 𝐴1 : N𝐺 [ 𝐴′ ] [𝑣] ⊆ 𝐴1

and

 𝐴1′′ = 𝑣 ∈ 𝐴1 : N𝐺 [ 𝐴′ ] [𝑣] ⊆ 𝐴1′ .

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

214

We note that 𝐴1′ ⊆ 𝐴′′ and 𝐴1 ⊆ 𝐴′ . If 𝑣 ∈ 𝐴1′ , then 𝑣 and all its neighbors belong to the set 𝐴′ , and so deg𝐺 (𝑣) = deg𝐺 [ 𝐴′ ] (𝑣). Thus, 𝐺 [ 𝐴′ ] is a graph with minimum degree 𝛿 > 1. Applying the inductive hypothesis to the graph 𝐺 [ 𝐴′ ], there is a subset 𝑆 𝑠 ⊆ 𝐴1′ such that 𝑆 𝑠 dominates 𝐴1′′ and |𝑆 𝑠 | ≤ 𝑓 (𝑠 − 1)| 𝐴1′ |. We now let 𝑆 𝑠+1 = ( 𝐴1 \ 𝐴1′ ) ∪ 𝑆 𝑠 ∪ 𝐵1 , and note that 𝑆 𝑠+1 ⊆ 𝐴′ and the set 𝑆 𝑠+1 dominates 𝐴′′ . Moreover,
E | 𝐴1 | = | 𝐴′ | 𝑝,
E |𝐵1 | ≤ | 𝐴′ |(1 − 𝑝) 1+ 𝛿 , and
E | 𝐴1′ | ≥ | 𝐴′ | 𝑝 1+Δ . Thus, by linearity of expectation,

E |𝑆 𝑠+1 | = E |( 𝐴1 \ 𝐴1′ ) ∪ 𝑆 𝑠 ∪ 𝐵1 |



= E | 𝐴1 | − E | 𝐴1′ | + E |𝑆 𝑠 | + E |𝐵1 |

≤ | 𝐴′ | 𝑝 + | 𝐴′ |(1 − 𝑝) 1+ 𝛿 − E | 𝐴1′ | + 𝑓 (𝑠 − 1)E | 𝐴1′ |

= | 𝐴′ | 𝑝 + | 𝐴′ |(1 − 𝑝) 1+ 𝛿 − 1 − 𝑓 (𝑠 − 1) E | 𝐴1′ |
≤ | 𝐴′ | 𝑝 + | 𝐴′ |(1 − 𝑝) 1+ 𝛿 − 1 − 𝑓 (𝑠 − 1) | 𝐴′ | 𝑝 1+Δ . By definition of the function 𝑓 ,   𝑠−1 ∑︁
1+Δ 1+ 𝛿 𝑖 (Δ+1) 1+ 𝛿
𝑝 1+Δ 𝑝 = 1 − 𝑝 − (1 − 𝑝) + 1 − 𝑝 − (1 − 𝑝) 1 − 𝑓 (𝑠 − 1) 𝑝 𝑖=1

= 1 − 𝑝 − (1 − 𝑝)

1+ 𝛿


𝑠 ∑︁

𝑝

𝑖 (Δ+1)

.

𝑖=1 (1− 𝑓 (𝑠−1) ) | 𝐴′ | 𝑝 1+Δ

Thus, z


′

′

E |𝑆 𝑠+1 | ≤ | 𝐴 | 𝑝 + | 𝐴 |(1 − 𝑝)

1+ 𝛿

′

}|

− | 𝐴 | 1 − 𝑝 − (1 − 𝑝)

1+ 𝛿


𝑠 ∑︁

{ 𝑝

𝑖 (Δ+1)

𝑖=1

  𝑠
∑︁ = | 𝐴′ | 𝑝 + (1 − 𝑝) 1+ 𝛿 − 1 − 𝑝 − (1 − 𝑝) 1+ 𝛿 𝑝 𝑖 (Δ+1) 𝑖=1

= 𝑓 (𝑠)| 𝐴′ |. Since expectation is an average value, there is a subset 𝑆 ⊆ 𝐴′ such that 𝑆 dominates 𝐴′′ and |𝑆| ≤ 𝑓 (𝑠)| 𝐴′ |. Arguments similar to those used in the proof of Lemma 7.9, together with the well-known inequality 1 − 𝑥 ≤ 𝑒 − 𝑥 for 0 ≤ 𝑥 ≤ 1, yield the following result. Lemma 7.10 ([509]) Let 𝐺 be a graph of order 𝑛 with minimum degree 𝛿 > 1 and maximum degree Δ, and let 0 < 𝑝 < 1. Let the sets 𝐴, 𝐴′ , and 𝐴′′ be defined as in

Section 7.2. Probabilistic Bounds

215

the statement of Lemma 7.9. For any integer 𝑠 ≥ 1, there is a subset 𝑆 ⊆ 𝐴′ such that 𝑆 dominates 𝐴′′ and |𝑆| ≤ 𝑓 (𝑠 − 1)| 𝐴′ |, where 𝑓 (0) = 𝑝 + 𝑒 − 𝑝 (1+ 𝛿 ) , and 𝑓 (0)

1− 𝑓 (0)

}| { ∑︁ z }| { z 𝑗 − 𝑝 (1+ 𝛿 ) − 𝑝 (1+ 𝛿 )
𝑓 ( 𝑗) = 𝑝 + 𝑒 𝑝 𝑖 (Δ+1) − 1− 𝑝−𝑒 𝑖=1

for every integer 𝑗 ≥ 1. We are now in a position to state and prove the result by Jafari Rad [509]. Theorem 7.11 ([509]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 > 1 and maximum degree Δ, then for every integer 𝑘 ≥ 1,  𝑖 (1+Δ) ! 𝑘 
∑︁ ln(𝛿 + 1) 𝑛 𝛾(𝐺) ≤ ln(𝛿 + 1) + 1 − 𝛿 − ln(𝛿 + 1) . 𝛿 + 1 𝛿 + 1 𝑖=1 Proof Let 𝐺 be a graph of order 𝑛 with minimum degree 𝛿 > 1 and maximum degree Δ, and let 𝑘 ≥ 1 be an integer. Let 𝐴 be a subset of vertices of 𝐺, where a vertex is chosen to be in 𝐴 with probability 𝑝 and independently of the choice for any other vertex, where ln(𝛿 + 1) 𝑝= . 𝛿+1 Let 𝐵 = 𝑉 \ N[ 𝐴] and let 𝐴′′ ⊆ 𝐴′ ⊆ 𝐴 be defined by   𝐴′ = 𝑣 ∈ 𝑉 : N[𝑣] ⊆ 𝐴 and 𝐴′′ = 𝑣 ∈ 𝑉 : N[𝑣] ⊆ 𝐴′ . By definition, if 𝑣 ∈ 𝐴′′ , then 𝑣 and its neighbors belong to the set 𝐴′ , and so deg𝐺 (𝑣) = deg𝐺 [ 𝐴′ ] (𝑣). Further, by definition, every vertex in 𝐴′ \ 𝐴′′ has a neighbor
in 𝐴 \ 𝐴′ . For 𝑝 = ln(𝛿 + 1) /(𝛿 + 1), Lemma 7.9 shows that there is a subset 𝑆 ⊆ 𝐴′ such that 𝑆 dominates 𝐴′′ and |𝑆| ≤ 𝑓 (𝑘 − 1)| 𝐴′ |, where 𝑓 (0) =

ln(𝛿 + 1) + 1 , and 𝛿+1

!  𝑗 
∑︁ ln(𝛿 + 1) 𝑖 (1+Δ) 1 𝑓 ( 𝑗) = ln(𝛿 + 1) + 1 − 𝛿 − ln(𝛿 + 1) 𝛿+1 𝛿+1 𝑖=1 for every integer 𝑗 ≥ 1. We now let 𝐷 = ( 𝐴 \ 𝐴′ ) ∪ 𝐵 ∪ 𝑆 and note that 𝐷 is a dominating set of 𝐺 satisfying |𝐷| = | 𝐴 \ 𝐴′ | + |𝐵| + |𝑆| ≤ | 𝐴| + |𝐵| − | 𝐴′ | + 𝑓 (𝑘 − 1)| 𝐴′ |
= | 𝐴| + |𝐵| − 1 − 𝑓 (𝑘 − 1) | 𝐴′ |.

216

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

By linearity of expectation,




E |𝐷 | ≤ E | 𝐴| + E |𝐵| − 1 − 𝑓 (𝑘 − 1) E | 𝐴′ | . Moreover,
E | 𝐴| = 𝑛𝑝,
E |𝐵| ≤ 𝑛(1 − 𝑝) 1+ 𝛿 , and
E | 𝐴′ | ≥ 𝑛𝑝 1+Δ . Thus, by linearity of expectation,




E |𝐷| ≤ E | 𝐴| + E |𝐵| − 1 − 𝑓 (𝑘 − 1) E | 𝐴′ |
≤ 𝑛𝑝 + 𝑛(1 − 𝑝) 1+ 𝛿 − 𝑛 1 − 𝑓 (𝑘 − 1) 𝑝 1+Δ
≤ 𝑛𝑝 + 𝑛𝑒 − 𝑝 (1+ 𝛿 ) − 1 − 𝑓 (𝑘 − 1) 𝑛𝑝 1+Δ

𝑛 ≤ ln(𝛿 + 1) + 1 − 𝑛 1 − 𝑓 (𝑘 − 1) 𝑝 1+Δ 𝛿+1  𝑖 (1+Δ) ! 𝑘 
∑︁ ln(𝛿 + 1) 𝑛 = ln(𝛿 + 1) + 1 − 𝛿 − ln(𝛿 + 1) . 𝛿 + 1 𝛿 + 1 𝑖=1 Since expectation is an average value, there is a dominating set 𝐷 ★ of 𝐺 of the desired cardinality. We note that the bound in Theorem 7.11 is better than the bound given in Theorem 7.2 for all 𝛿 ≥ 2, since 𝛿 − ln(𝛿 + 1) > 0. Theorem 7.12 ([509]) If 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 > 1 and maximum degree Δ, then for every integer 𝑘 ≥ 1,  𝑖 (1+Δ) ! 𝑘  ∑︁ 𝛿 𝛿 1 𝛾(𝐺) ≤ 1 − − 1− 𝑛. 1 1 1 (𝛿 + 1) 1+ 𝛿 (𝛿 + 1) 1+ 𝛿 𝑖=1 (𝛿 + 1) 𝛿 Proof The proof is analogous to the proof given for Theorem 7.11, except that in this case we apply Lemma 7.10 with 𝑝 =1−

1

, 1

(𝛿 + 1) 𝛿

and define 𝑓 (0) = 1 −

𝛿 1

(𝛿 + 1) 1+ 𝛿

, and 1− 𝑓 (0)

𝑓 (0)

}| 𝛿

z 𝑓 ( 𝑗) = 1 −

{ z 1

(𝛿 + 1) 1+ 𝛿

for every integer 𝑗 ≥ 1.

−

}| 𝛿

{

𝑗  ∑︁

1

(𝛿 + 1) 1+ 𝛿

𝑖=1

1−

 𝑖 (1+Δ)

1 1

(𝛿 + 1) 𝛿

Section 7.3. Domination in Random Graphs

217

We note that the bound in Theorem 7.12 is better than the bound given in Theorem 7.3 for all 𝛿 ≥ 2. We also note that if 𝐺 is a 𝛿-regular graph, then the bound in Theorem 7.12 is better than the bound given in Theorem 7.8 for all 𝛿 ≥ 2. Similar probabilistic bounds also exist for the total domination number. Recall that in Theorem 6.77 in Chapter 6, we proved that if 𝐺 is a graph of order 𝑛 with minimum degree 𝛿 ≥ 1, then   1 + ln(𝛿) 𝛾t (𝐺) ≤ 𝑛. 𝛿 This bound can also be proved using probabilistic arguments analogous to the probabilistic proof of Theorem 7.2. In 2019 Henning and Jafari Rad [462] gave the following improved probabilistic upper bound on the total domination number of a graph in terms of its minimum degree. Theorem 7.13 ([462]) If 𝐺 is a graph with minimum degree 𝛿 ≥ 2 and maximum degree Δ, then     1   1+Δ(Δ−1) 1 + ln(𝛿) 1 1 𝛿−1 ln(𝛿) 𝛾t (𝐺) ≤ 𝑛 −𝑛 1− . 𝛿 𝛿 Δ 𝛿 

7.3

Domination in Random Graphs

In this section, we study the behavior of the domination, total domination, and independent domination numbers using the Erdős-Rényi random graph model, which they introduced in [258] in 1960. For a positive integer 𝑛 and real number 𝑝 with 0 < 𝑝 < 1, we denote by G(𝑛, 𝑝) the probability space whose elements 𝐺 are all possible graphs of order 𝑛 with vertex set 𝑉𝑛 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 }, where an edge is chosen to be in 𝐺 with probability 𝑝 and independently of the choice for any other edge. We call an element 𝐺 ∈ G(𝑛, 𝑝) a random graph. A graph property or graph invariant is a property of graphs that is preserved under all possible isomorphisms of a graph, and therefore depends only on the abstract structure of graphs and not on the way the graph is drawn or represented. Two examples of graph properties are connectedness and having diameter 2. Let P be the set of graphs of order 𝑛 having Property 𝑃, and let 𝐺 ∈ G(𝑛, 𝑝) be a random graph, where 0 < 𝑝 < 1 and where 𝑝 may depend on 𝑛. If lim Pr(𝐺 ∈ P) = 1,

𝑛→∞

then we say that almost every random graph 𝐺 ∈ G(𝑛, 𝑝) has property 𝑃, or that the property 𝑃 holds asymptotically almost surely (abbreviated, a.a.s.). On the other hand, if lim Pr(𝐺 ∈ P) = 0, 𝑛→∞

then we say that almost no random graph 𝐺 ∈ G(𝑛, 𝑝) has property 𝑃. In 1981 Weber [747] was the first to study the domination and independent domination numbers of random graphs.

218

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

Theorem 7.14 ([747]) If 𝐺 ∈ G(𝑛, 𝑝) is a random graph of order 𝑛 with 𝑝 = 21 , and 
 𝜉1 (𝑛) = log2 (𝑛) − log2 log2 (𝑛) ln(𝑛) , then a.a.s. 𝛾(𝐺) = 𝜉1 (𝑛) + 1 or 𝛾(𝐺) = 𝜉1 (𝑛) + 2, and 𝑖(𝐺) = 𝜉1 (𝑛) + 2 or 𝑖(𝐺) = 𝜉1 (𝑛) + 3. In 2001 Wieland and Godbole [751] extended Weber’s result and showed that the domination number of a random graph has the following two-point concentration. Theorem 7.15 ([751]) Let 𝐺 ∈ G(𝑛, 𝑝) be a random graph of order 𝑛, where either 𝑝 is a constant or 𝑝 = 𝑝(𝑛) such that  2  ln (𝑛) 2 𝑝 ln(𝑛) ≥ 40 ln . 𝑝 If 
 𝜉2 (𝑛) = log𝑞 (𝑛) − log𝑞 log𝑞 (𝑛) ln(𝑛) , where 𝑞 =

1 1− 𝑝

and log𝑞 denotes the logarithm with base 𝑞, then a.a.s. 𝛾(𝐺) = 𝜉2 (𝑛) + 1 or

𝛾(𝐺) = 𝜉2 (𝑛) + 2.

We remark that the probability function 𝑝 = 𝑝(𝑛) in the statement of Theorem 7.15 tends to 0 as 𝑛 tends to infinity and is equivalent (see [339]) to √︄
10 ln ln(𝑛) 𝑝 = 𝑝(𝑛) ≥ . ln(𝑛) The proof of Theorem 7.15, which we omit, uses first and second moment methods to prove the two-point concentration of the domination number. Wieland and Godbole [751] raised the problem whether the validity of this two-point concentration in Theorem 7.15 can be extended to a wider range of 𝑝. This question was answered in 2015 by Glebov et al. [339]. Theorem 7.16 ([339])

If 𝐺 ∈ G(𝑛, 𝑝) is a random graph of order 𝑛 with ln(𝑛) √ ≪ 𝑝 < 1, 𝑛

and  𝜉3 (𝑛, 𝑝) = log𝑞 where 𝑞 =

1 1− 𝑝 ,



𝑛 ln(𝑞) ln2 (𝑛𝑝)

 (1 + O (1) ,

then a.a.s. 𝛾(𝐺) = 𝜉3 (𝑛, 𝑝) + 1 or

𝛾(𝐺) = 𝜉3 (𝑛, 𝑝) + 2.

In 2014 Henning and Yeo [491] showed that if we choose the probability 𝑝 carefully, then the domination and total domination numbers of the random graph G(𝑛, 𝑝)

Section 7.3. Domination in Random Graphs

219

√︁ √︁ are concentrated between roughly √1 𝑛 ln(𝑛) and √1 𝑛 ln(𝑛). Before we present 2 2 2 a precise statement and proof of this result, we first give some preliminary results. We shall need the following well-known lemma. Lemma 7.17 For all 𝑥 > 1, 1−


1 𝑥 𝑥

< 𝑒 −1 < 1 −


1 𝑥−1 . 𝑥

In what follows, we define √︄ 𝑐𝑛 =


ln ln(𝑛) 1 1 . + + 2 ln(𝑛) 2 ln(𝑛)

The function 1/ln(𝑛) is a decreasing function. For 𝑛 ≥ 24, we therefore have 1/ln(𝑛) ≤ 1/ln(24). Using the well-known calculus result that for 𝑥 > 0 the maximum
value of ln(𝑥)/𝑥 is 𝑒 −1 , we have that ln ln(𝑛) /ln(𝑛) ≤ 𝑒 −1 . Hence, √︄ 1 1 1 𝑐𝑛 ≤ + + < 1. 2 ln(24) 2𝑒 We state this formally as follows. Observation 7.18 If 𝑛 ≥ 24, then 𝑐 𝑛 < 1. We shall need the following upper bound on the total domination number of a graph of diameter 2. Lemma 7.19 ([491]) If 𝐺 is a diameter-2 graph of order 𝑛 ≥ 24, then 𝛾t (𝐺) < √︁ 𝑐 𝑛 𝑛 ln(𝑛) + 1. Proof Let 𝐺 be a diameter-2 graph of order 𝑛 ≥ 24. Since the closed neighborhood of any vertex in a diameter-2 graph is a TD-set in the graph, we can choose a vertex of minimum degree 𝛿(𝐺) =√︁𝛿 and form a TD-set of cardinality 𝛿 + 1. Hence, 𝛾t (𝐺) ≤ 𝛿 + 1. Thus, if 𝛿 < 𝑐 𝑛 𝑛 ln(𝑛), then the desired result is immediate. √︁ Therefore, we may assume that 𝛿 ≥ 𝑐 𝑛 𝑛 ln(𝑛). Since 𝑛 ≥ 24, by Observation 7.18, we have 𝑐 𝑛 < 1,
implying that ln(𝑐 𝑛 ) < 0. Thus, by Theorem 6.77 in Chapter 6 and since 1 + ln(𝛿) /𝛿 is a decreasing function for all 𝛿 ≥ 1, the following holds: √︁
    1 + ln 𝑐 𝑛 𝑛 ln(𝑛) 1 + ln(𝛿) 𝛾t (𝐺) ≤ 𝑛≤ 𝑛 √︁ 𝛿 𝑐 𝑛 𝑛 ln(𝑛)
 √︁  𝑛 ln(𝑛) ln(𝑛) ln ln(𝑛) = 1 + ln(𝑐 𝑛 ) + + 2 2 𝑐 𝑛 ln(𝑛)
  √︁ ln ln(𝑛) 1 1 1 < + + 𝑛 ln(𝑛) 𝑐 𝑛 ln(𝑛) 2 2 ln(𝑛) √︁ 1 = × 𝑐2𝑛 × 𝑛 ln(𝑛) 𝑐𝑛 √︁ = 𝑐 𝑛 𝑛 ln(𝑛).

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

220

As a consequence of Lemma 7.19, we have the following result. Theorem 7.20 ([491]) Given any 𝜀 > 0, if 𝐺 is a diameter-2 graph of sufficiently large order 𝑛, then   √︁ 1 𝛾t (𝐺) < √ + 𝜀 𝑛 ln(𝑛). 2 Proof The function √︄

tends to holds:

√1 2


ln ln(𝑛) 1 1 1 + √︁ + + , 2 ln(𝑛) 2 ln(𝑛) 𝑛 ln(𝑛)

when 𝑛 tends to infinity. Therefore, for 𝑛 sufficiently large, the following √︄


ln ln(𝑛) 1 1 1 + √︁ + + < 2 ln(𝑛) 2 ln(𝑛) 𝑛 ln(𝑛)
√︄  √︁ ln ln(𝑛) 1 1 × 𝑛 ln(𝑛) + 1 < + + 2 ln(𝑛) 2 ln(𝑛) √︁ 𝑐 𝑛 𝑛 ln(𝑛) + 1
2 2 𝜀 > 𝜀 > 𝜀 ′ . Let 𝜀1 be defined such 2 2 that 0 < 𝜀 ′ < 𝜀1 < 𝜀★, and let 𝑁 be large enough so that

Section 7.3. Domination in Random Graphs 

1 1 + 𝜀1

 √︂

221

  √︂ 𝑛 ln(𝑛) 1 𝑛 ln(𝑛) − >1 8 1 + 𝜀★ 8

holds for all 𝑛 > 𝑁. Let  𝑡=

1 1 + 𝜀★

 √︂

𝑛 ln(𝑛) 8



and define 𝜀★ 1 such that 

1 𝑡= 1 + 𝜀★ 1

 √︂

𝑛 ln(𝑛) . 8

★ When 𝑛 > 𝑁, we note that 0 < 𝜀 ′ < 𝜀 1 < 𝜀★ 1 ≤ 𝜀 . By definition of 𝑝 and 𝑡, and ★ since 𝜀1 < 𝜀1 , (1 + 𝜀 ′ ) ln(𝑛) (1 + 𝜀 ′ ) ln(𝑛) 𝑝𝑡 = < . 2(1 + 𝜀1 ) 2(1 + 𝜀★ 1)

For any 𝜀2 such that 0 < 𝜀2 < 1, let 𝑁 be large enough so that the following holds when 𝑛 > 𝑁. Then   √︂    √︂  1 𝑛 ln(𝑛) 1 ln(𝑛) 𝑛−𝑡 =𝑛− ≥ 𝑛 1 − ≥ 𝑛(1 − 𝜀2 ). 8 1 + 𝜀1 8𝑛 1 + 𝜀★ 1 Let 𝜀3 be defined such that 0 < 𝜀3 < 1 −

1 + 𝜀′ , 1 + 𝜀1

which is possible as 𝜀 ′ < 𝜀 1 . Let 𝑁 be large enough so that 1 − 𝑝 ≥ 1 − 𝜀3 when 𝑛 > 𝑁, which is possible as 𝑝 tends to zero when 𝑛 goes to infinity. Let 𝜀4 = 1 −

1 + 𝜀′ (1 + 𝜀1 ) (1 − 𝜀3 )

and note that by the definition of 𝜀3 , we have 𝜀4 > 0. Further, we note that none ★ ′ ★ of 𝜀★, 𝜀 ′ , 𝜀1 , 𝜀2 , 𝜀3 , and 𝜀4 depends on 𝑛 (but 𝜀★ 1 does). The properties of 𝜀 , 𝜀 , 𝜀 1 , 𝜀1 , 𝜀2 , 𝜀3 , and 𝜀4 when 𝑛 > 𝑁 are summarized in the following claim. Claim 7.21.1 The following hold: ★ (a) 0 < 𝜀 ′ < 𝜀1 < 𝜀★ 1 ≤ 𝜀 . (b) 𝑛 − 𝑡 ≥ 𝑛(1 − 𝜀2 ) and 0 < 𝜀2 < 1. (c) 1 − 𝑝 ≥ 1 − 𝜀3 . (d) 𝜀4 > 0. ′ ) ln(𝑛) (e) 𝑝 × 𝑡 < (1+𝜀 2(1+𝜀1 ) . We are now in a position to establish the lower bound in the statement of the theorem.
√︁ Claim 7.21.2 If 𝐺 ∈ G(𝑛, 𝑝), then a.a.s. 𝛾(𝐺) > √1 − 𝜀 𝑛 ln(𝑛). 2 2

222

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

Proof Let 𝐺 ∈ G(𝑛, 𝑝) and let 𝑉 = 𝑉 (𝐻) = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 }. Let 𝑆 be an arbitrary subset of vertices of 𝐺 with |𝑆| = 𝑡. Let 𝑥 ∈ 𝑉 \ 𝑆 be arbitrary. The probability that 𝑆 does not dominate 𝑥 is (1 − 𝑝) 𝑡 , since there are 𝑡 vertices in 𝑆 and each vertex has probability of 1 − 𝑝 of not being adjacent to 𝑥. Therefore, vertex 𝑥 has probability 1 − (1 − 𝑝) 𝑡 of being dominated by 𝑆. The probability of one vertex in 𝑉 \ 𝑆 being dominated by 𝑆 is independent of any other vertex in 𝑉 \ 𝑆 being dominated by 𝑆. Thus, the probability that 𝑆 dominates all of 𝑉 \ 𝑆 is given by
𝑛−𝑡 (7.1) Pr(𝑆 is a dominating set) = 1 − (1 − 𝑝) 𝑡
𝑛−𝑡 since |𝑉 \𝑆| = 𝑛−𝑡. We now provide an upper bound on the expression 1−(1− 𝑝) 𝑡 as follows. We first establish the following lower bound on the term (1 − 𝑝) 𝑡 :
( 1 −1) 𝑝𝑡 (1 − 𝑝) 𝑡 = 1 − 1/1𝑝 𝑝 1− 𝑝
𝑝𝑡 > 𝑒 −1 1− 𝑝 (by Lemma 7.17) 𝑝𝑡
≥ 𝑒 −1 1− 𝜀3 (by Claim 7.21.1(c)) (1+𝜀 ′ ) ln(𝑛)
≥ 𝑒 −1 2(1+𝜀1 ) (1− 𝜀3 ) (by Claim 7.21.1(d))
(1− 𝜀4 ) ln(𝑛)/2 = 𝑒 −1 (by definition of 𝜀4 ), and so, (1 − 𝑝) 𝑡 ≥ 𝑛 − (1− 𝜀4 )/2 . (7.2) Hence, the probability that 𝑆 is a dominating set is bounded as follows:
𝑛−𝑡 Pr(𝑆 is a dominating set) = 1 − (1 − 𝑝) 𝑡 (by Equation (7.1))   𝑛 [ (1− 𝜀4 ) /2] × [ (1−𝑛−𝑡 𝜀4 ) /2] 𝑛 1 ≤ 1 − (1− 𝜀 )/2 (by Inequality (7.2)) 4 𝑛
𝑛−𝑡 < 𝑒 −1 𝑛 [ (1− 𝜀4 ) /2] (by Lemma 7.17) 𝑛(1− 𝜀2 )
≤ 𝑒 −1 𝑛 [ (1− 𝜀4 ) /2] (by Claim 7.21.1(b)) [ (1+𝜀4 ) /2]
(1− 𝜀 )𝑛 2 = 𝑒 −1 . 𝑛
As there are 𝑡 possible choices for choosing 𝑆, the probability that 𝛾(𝐺) ≤ √ 𝑛 ln(𝑛) √ is bounded as follows: ★

2 2(1+𝜀 )



√︁





√︁

 𝑛 ln(𝑛) Pr 𝛾(𝐺) ≤ √ ≤ Pr 𝛾(𝐺) ≤ √ 2 2(1 + 𝜀★) 2 2(1 + 𝜀★ 1)   𝑛 ≤ × Pr(𝑆 is a dominating set) 𝑡
(1− 𝜀2 )𝑛 [ (1+𝜀4 ) /2] ≤ 𝑛𝑡 × 𝑒 −1 𝑛 ln(𝑛)

= 𝑒𝑡

ln(𝑛) − (1− 𝜀2 )𝑛 [ (1+𝜀4 ) /2]

.

Section 7.3. Domination in Random Graphs

223

By the definition of 𝑡, it follows that 𝑡 ln(𝑛) − (1 − 𝜀2 )𝑛

[ (1+𝜀4 )/2]

√︂  1 𝑛 ln(𝑛) = ln(𝑛) − (1 − 𝜀2 )𝑛 [ (1+𝜀4 )/2] 8 1 + 𝜀★ 1 √︁   √ ln(𝑛) ln(𝑛) ≤ √ − (1 − 𝜀2 )𝑛 𝜀4 /2 𝑛. (1 + 𝜀1 ) 8



By Claim 7.21.1, we have that 𝜀1 > 0, 0 < 𝜀2 < 1, and 𝜀4 > 0. Since 𝑛 𝜀4 /2 grows faster than any poly-log function in 𝑛, we have that 𝑡 ln(𝑛) − (1 − 𝜀2 )𝑛 (1+𝜀4 )/2 → −∞ as 𝑛 → ∞, implying that √︁



𝑛 ln(𝑛)



Pr 𝛾(𝐺) ≤ √ →0 2 2(1 + 𝜀★) as 𝑛 → ∞. Hence, a.a.s. we have √︁

  √︁ 𝑛 ln(𝑛) 1 𝛾(𝐺) > √ = √ − 𝜀 𝑛 ln(𝑛), 2 2(1 + 𝜀★) 2 2 which completes the proof of Claim 7.21.2. We prove next that a.a.s. if 𝐺 ∈ G(𝑛, 𝑝), then diam(𝐺) = 2. For this purpose, we define a bad pair of vertices in a graph 𝐺 to be a pair of vertices 𝑢 and 𝑣 at distance at least 3 in 𝐺. In particular, if 𝑢 and 𝑣 is a bad pair, then 𝑢 and 𝑣 are neither adjacent nor do they have a common neighbor. We note that diam(𝐺) > 2 if and only if 𝐺 has a bad pair of vertices. Claim 7.21.3 If 𝐺 ∈ G(𝑛, 𝑝), then a.a.s. 𝐺 is a diameter-2 graph. Proof Let 𝐺 ∈ G(𝑛, 𝑝) with 𝑉 = 𝑉 (𝐺) and let 𝑢 and 𝑣 be an arbitrary pair of bad vertices in the graph 𝐺 and let 𝑤 ∈ 𝑉 \ {𝑢, 𝑣}. The probability that 𝑤 is adjacent to both 𝑢 and 𝑣 is 𝑝 2 and hence the probability that 𝑤 is not a common neighbor of 𝑢 and 𝑣 is 1 − 𝑝 2 . This is true for all 𝑛 − 2 vertices in 𝑉 \ {𝑢, 𝑣}. Hence, the
𝑛−2 probability that no vertex is adjacent to both 𝑢 and 𝑣 is 1 − 𝑝 2 . Further, the probability that 𝑢 and 𝑣 are not adjacent is 1 − 𝑝. Hence, the probability that 𝑢
𝑛−2 and 𝑣 is a bad pair is (1 − 𝑝) 1 − 𝑝 2 . When 𝑛 is sufficiently large, we note that √

2 𝑝 ≤ −1 + 5 /2 < 0.618 and therefore that (1 − 𝑝)/ 1 − 𝑝 2 < 1. Hence, for 𝑛 large enough, the following holds by Lemma 7.17,

224

Chapter 7. Probabilistic Bounds and Domination in Random Graphs
𝑛−2
Pr 𝑑 (𝑢, 𝑣) > 2 = (1 − 𝑝) 1 − 𝑝 2   12 × 𝑝2 𝑛 𝑝 1− 𝑝 1 = 1 − 2 2 2 (1 − 𝑝 ) 1/𝑝   12 × 𝑝2 𝑛 𝑝 1 < 1− 2 1/𝑝
𝑝2 𝑛 < 𝑒 −1
(1+𝜀 ′ ) 2 × ( 2 ln(𝑛) 𝑛 )𝑛 = 𝑒 −1 1 = . ′ )2 2(1+𝜀 𝑛

Let 𝑋 be
the random variable that counts the number of bad pairs in 𝐺. Since there are 𝑛2 pairs of vertices, the expected number of bad pairs is E(𝑋) =

∑︁

Pr 𝑑 (𝑢, 𝑣, >)2




{𝑢,𝑣 } ⊆𝑉

   𝑛 1 2 𝑛2(1+𝜀 ′ ) 2   1 ≤ 𝑛2 𝑛2(1+2𝜀 ′ +( 𝜀 ′ ) 2 ) 1 = . ′ +2( 𝜀 ′ ) 2 4𝜀 𝑛 ≤

Thus, lim E(𝑋) → 0,

𝑛→∞

implying that a.a.s. the graph 𝐺 has no bad pairs and √︁ hence has diameter at most 2. By Claim 7.21.2, a.a.s. we have 𝛾(𝐺) > √ 1 ★ 𝑛 ln(𝑛) > 1, implying that 𝐺 2 2(1+𝜀 ) does not have a vertex adjacent to every other vertex. Therefore, a.a.s. we have diam(𝐺) = 2. The upper bound now follows from Claim 7.21.3 and Theorem 7.20. In 2021 Dubickas [245] posed the following problem. Problem 7.22 ([245]) Find 𝑐 min = lim inf 𝑛→∞

min𝐺 ∈ G𝑛 𝛾t (𝐺) √︁ 𝑛 ln(𝑛)

and

𝑐 max = lim sup 𝑛→∞

min𝐺 ∈ G𝑛 𝛾t (𝐺) , √︁ 𝑛 ln(𝑛)

where G𝑛 is the class of diameter-2 graphs of order 𝑛. In 2014 Desormeaux et al. [230] showed that 14 ≤ 𝑐 min . As a consequence of Theorem 7.21, we have the improved result that √1 ≤ 𝑐 min and 𝑐 max ≤ √1 . 2 2

2

Section 7.3. Domination in Random Graphs

225

√︃ Dubickas [245] further improved the value of 𝑐 min to 38 ≤ 𝑐 min . Thus, the best results to date are given by √︃ 0.612372 ≈ 38 ≤ 𝑐 min ≤ 𝑐 max ≤ √1 ≈ 0.707107. 2

The exact values of 𝑐 min and 𝑐 max have yet to be determined. Dubickas [245] √︃ 3 wrote: “It is quite tempting to conjecture that 𝑐 min = 𝑐 max = 8 .” Results similar to those for the domination number and total domination number also hold for the independent domination number of a random graph. In 2008 Bonato and Wang [85] proved the following two-point concentration result. Theorem 7.23 ([85]) If 𝐺 ∈ G(𝑛, 𝑝) is a random graph of order 𝑛 with 𝑝 a constant, then a.a.s. 
 log𝑞 (𝑛) − log𝑞 log𝑞 (𝑛) ln(𝑛) + 1 ≤ 𝑖(𝐺) ≤ ⌊log𝑞 (𝑛)⌋, where 𝑞 =

1 1− 𝑝 .

The result of Theorem 7.23 was improved in 2011 by Clark and Johnson [179], who proved the following two-point concentration for the independent domination number of a random graph. Theorem 7.24 ([179]) Let 𝐺 ∈ G(𝑛, 𝑝) be a random graph of order 𝑛, where either 𝑝 is a constant or 𝑝 = 𝑝(𝑛) such that   ln(𝑛) 2 𝑝 ln(𝑛) ≥ 64 ln . 𝑝 If  
𝜉4 (𝑛) = log𝑞 (𝑛) − log𝑞 log𝑞 (𝑛) ln(𝑛) + log𝑞 (2) , where 𝑞 =

1 1− 𝑝

then a.a.s. 𝑖(𝐺) = 𝜉4 (𝑛) + 1 or 𝑖(𝐺) = 𝜉4 (𝑛) + 2.

Theorem 7.24 extends the result of Weber in Theorem 7.23 in the case when 𝑝 = 12 , and the result of Bonato and Wang in Theorem 7.23 in the case when 𝑝 is constant. We close this section with results on the domination and independent domination numbers of a random cubic graph. In 1995 Molloy and Reed [599] established the following bounds on the domination number of a random cubic graph. Theorem 7.25 ([599]) If 𝐺 is a random cubic graph of order 𝑛, then a.a.s. 0.2636𝑛 ≤ 𝛾(𝐺) ≤ 0.3126𝑛. In 2002 Duckworth and Wormald [246] proved a better upper bound for the independent domination number 𝑖(𝐺) of a random cubic graph than the upper bound for 𝛾(𝐺) in Theorem 7.25.

226

Chapter 7. Probabilistic Bounds and Domination in Random Graphs

Theorem 7.26 ([246]) If 𝐺 is a random cubic graph of order 𝑛, then a.a.s. 0.2641𝑛 ≤ 𝑖(𝐺) ≤ 0.27942𝑛. The lower bound in Theorem 7.26 was calculated by means of a direct expectation argument. The upper bound can be calculated by using differential equations to analyze the performance of a randomized greedy algorithm used in [246]. A simple randomized greedy algorithm for finding an independent dominating set 𝑆 in an arbitrary graph 𝐺 can be constructed by iteratively selecting a vertex 𝑣 to be in 𝑆 and then removing 𝑣 and all its neighbors N[𝑣] from the graph 𝐺. In the remaining graph, select a second vertex for the set 𝑆 and then remove it and all of its neighbors. Repeat this process until there are no remaining vertices. Removing a selected vertex and its neighbors guarantees that the final set is independent and all vertices removed are either in the set 𝑆 or are dominated by a vertex in 𝑆. Thus, when no vertices remain, 𝑆 is guaranteed to be an independent dominating set. This algorithm is highlighted in Algorithm 4. The choice of the next vertex to be selected for the set 𝑆 can be made using a variety of heuristics, such as selecting a vertex of maximum or minimum degree in the remaining subgraph. Such a greedy algorithm becomes randomized when the vertex is selected uniformly at random from the set of vertices meeting the selection criteria. In [246] the authors selected a vertex of minimum degree. Algorithm 4 Randomized Independent Dominating Set Input : A graph 𝐺 = (𝑉, 𝐸) Output : An independent dominating set 𝑆 of 𝐺 1 2

3 4

5 6 7

[Initialize 𝑆 and the set 𝑊 from which vertices are to be selected] Set 𝑆 = ∅ Set 𝑊 = 𝑉 [Iteratively select vertices for 𝑆 and remove them and their neighbors] while 𝑊 ≠ ∅ do Select a vertex 𝑣 uniformly at random from 𝑊 of minimum degree in 𝐺 [𝑊] Set 𝑆 = 𝑆 ∪ {𝑣} Set 𝑊 = 𝑊 − N[𝑣] od

7.4

Summary

In this chapter, we presented probabilistic bounds for the domination and total domination numbers of a graph in terms of its order and minimum degree. We showed that the bounds are optimal when the minimum degree 𝛿 is sufficiently large. We also showed that if we carefully choose the probability 𝑝 that an each edge is chosen in a random graph G(𝑛, 𝑝) of order 𝑛, then the domination, total domination, and independent domination numbers enjoy a tight concentration.

Chapter 8

Bounds in Terms of Size 8.1 Introduction The order |𝑉 | = 𝑛 and size |𝐸 | = 𝑚 of a graph 𝐺 = (𝑉, 𝐸) are the two most fundamental graph parameters. In this chapter, we study how the size of a graph relates to its domination, total domination, and independent domination numbers.

8.2 Domination and Size One of the earliest results relating the domination number and the size of a graph is the following 1962 result due to Berge [67]. Theorem 8.1 ([67]) If 𝐺 is a graph of order 𝑛 and size 𝑚, then 𝛾(𝐺) ≥ 𝑛 − 𝑚, with equality if and only if each component of 𝐺 is a star. Proof Let 𝐺 have order 𝑛 and size 𝑚. Since 𝛾(𝐺) ≥ 1, the result is immediate for 𝑚 ≥ 𝑛. Hence, we may assume that 𝑚 ≤ 𝑛−1. Let 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑘 be the components of 𝐺 where 𝑘 ≥ 1, and let 𝐺 𝑖 have order 𝑛𝑖 and size 𝑚 𝑖 for 𝑖 ∈ [𝑘]. We note that 𝑚=

𝑘 ∑︁ 𝑖=1

𝑚𝑖 ≥

𝑘 ∑︁

(𝑛𝑖 − 1) = 𝑛 − 𝑘,

(8.1)

𝑖=1

with equality if and only if 𝐺 𝑖 is a tree for each 𝑖 ∈ [𝑘]. In particular, 𝑘 ≥ 𝑛 − 𝑚. By linearity, 𝑘 𝑘 ∑︁ ∑︁ 𝛾(𝐺) = 𝛾(𝐺 𝑖 ) ≥ 1 = 𝑘 ≥ 𝑛 − 𝑚. (8.2) 𝑖=1

𝑖=1

This establishes the lower bound. Further, if 𝛾(𝐺) = 𝑛 − 𝑚, then we have equality throughout Inequality (8.2), implying that 𝑘 = 𝑛 − 𝑚 and 𝛾(𝐺 𝑖 ) = 1 for all 𝑖 ∈ [𝑘]. This in turn implies that we have equality throughout Inequality (8.1), and so 𝐺 𝑖 is a tree with a vertex adjacent to every other vertex in 𝐺 𝑖 for all 𝑖 ∈ [𝑘], that is, either © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_8

227

Chapter 8. Bounds in Terms of Size

228

𝑛𝑖 = 1 in which case 𝐺 𝑖 = 𝐾1 , or 𝑛𝑖 ≥ 2 in which case 𝐺 𝑖 = 𝐾1,𝑛𝑖 −1 for all 𝑖 ∈ [𝑘]. Thus, every component of 𝐺 is a star. Conversely, if every component of 𝐺 is a star, then 𝛾(𝐺) = 𝑛 − 𝑚. In 1965 Vizing [733] proved a classic result bounding the size of a graph in terms of its order and domination number. Theorem 8.2 (Vizing’s Theorem [733]) If 𝐺 is a graph of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾, then 𝑚 ≤ 12 (𝑛 − 𝛾) (𝑛 − 𝛾 + 2). Proof We proceed by induction on the order 𝑛 of a graph of size 𝑚 with domination number 𝛾. The result is immediate if 𝑛 = 1, since in this case 𝑚 = 0 and 𝛾 = 1. This establishes the base case. Assume that 𝑛 ≥ 2 and that if 𝐺 ′ is a graph of order 𝑛′ and size 𝑚 ′ with domination number 𝛾 ′ , where 1 ≤ 𝑛′ < 𝑛, then 𝑚 ′ ≤ 12 (𝑛′ − 𝛾 ′ ) (𝑛′ − 𝛾 ′ + 2). Let 𝐺 be a graph of order 𝑛 and size 𝑚 with domination number 𝛾. Let Δ = Δ(𝐺).
If 𝛾 = 1, then 𝑚 ≤ 𝑛2 < 12 (𝑛 − 1)(𝑛 + 1) = 12 (𝑛 − 𝛾) (𝑛 − 𝛾 + 2). Hence, we may assume that 𝛾 ≥ 2. If 𝐺 is the empty graph 𝐾 𝑛 , then 𝛾 = 𝑛 and 𝑚 = 0 = (𝑛 − 𝛾) (𝑛 − 𝛾 + 2)/2. Hence, we may further assume that 𝑚 ≥ 1, and so Δ ≥ 1. By Theorem 4.3, we have 𝛾 ≤ 𝑛 − Δ. Let 𝑣 be a vertex of 𝐺 of maximum degree Δ, and so deg(𝑣) = |N(𝑣)| = Δ ≤ 𝑛 − 𝛾. Thus, |N(𝑣)| = 𝑛 − 𝛾 − 𝑘 for some integer 𝑘, where 0 ≤ 𝑘 ≤ 𝑛 − 𝛾 −1. Let 𝑆 = 𝑉 \N[𝑣], and so
|𝑆| = 𝛾 + 𝑘 −1. Let 𝑢 be an arbitrary neighbor of 𝑣, and consider the set 𝑆𝑢 = 𝑆 \N(𝑢) ∪ {𝑢, 𝑣}. The set 𝑆𝑢 is a dominating set of 𝐺 and so, 𝛾 ≤ |𝑆𝑢 | = |𝑆| − |𝑆 ∩ N(𝑢)| + 2 = (𝛾 + 𝑘 − 1) − |𝑆 ∩ N(𝑢)| + 2, or equivalently, |𝑆 ∩ N(𝑢)| ≤ 𝑘 + 1

for all 𝑢 ∈ N(𝑣).

(8.3)

Let 𝑚 1 be the number of edges between N(𝑣) and 𝑆, and so, 𝑚 1 = | [N(𝑣)]𝑆|. By Inequality (8.3), 𝑚 1 ≤ Δ(𝑘 + 1).

(8.4)

Let 𝐺 ′ = 𝐺 [𝑆] and let 𝐷 ′ be a 𝛾-set of 𝐺 ′ . Let 𝐺 ′ have order 𝑛′ , size 𝑚 ′ , and domination number 𝛾 ′ . Thus, 𝑛′ = |𝑆| = 𝛾 + 𝑘 − 1. The set 𝐷 ′ ∪ {𝑣} is a dominating set of 𝐺 and so, 𝛾 ≤ |𝐷 ′ | + 1. Thus, 𝛾 ′ = |𝐷 ′ | ≥ 𝛾 − 1 ≥ 1. We note that 𝑛′ − 𝛾 ′ ≤ (𝛾 + 𝑘 − 1) − (𝛾 − 1) = 𝑘. Applying the inductive hypothesis to the graph 𝐺 ′ , we have 𝑚 ′ ≤ 12 (𝑛′ − 𝛾 ′ ) (𝑛′ − 𝛾 ′ + 2) ≤ 12 𝑘 (𝑘 + 2).

(8.5)

Recall that deg(𝑣) = Δ and 𝑘 = 𝑛 − 𝛾 − |N(𝑣)| = 𝑛 − 𝛾 − Δ. We note that |N[𝑣] | = Δ+1 and each vertex in N[𝑣] has degree at most Δ in 𝐺. By Inequalities (8.4) and (8.5),

Section 8.2. Domination and Size 2𝑚 =

∑︁

deg(𝑢) +

𝑢∈N[𝑣 ]

∑︁

229 deg(𝑢)

𝑢∈𝑆 ′

≤ Δ(Δ + 1) + (2𝑚 + 𝑚 1 ) ≤ Δ(Δ + 1) + 𝑘 (𝑘 + 2) + Δ(𝑘 + 1) = Δ(Δ + 1) + (𝑛 − 𝛾 − Δ) (𝑛 − 𝛾 − Δ + 2) + Δ(𝑛 − 𝛾 − Δ + 1) = (𝑛 − 𝛾) (𝑛 − 𝛾 + 2) − Δ(𝑛 − 𝛾 − Δ) ≤ (𝑛 − 𝛾) (𝑛 − 𝛾 + 2), or equivalently, 𝑚 ≤ 21 (𝑛 − 𝛾) (𝑛 − 𝛾 + 2). As shown in the proof of Theorem 8.2, if 𝐺 is a graph of order 𝑛 and size 𝑚 with
domination number 𝛾 = 1, then 𝑛2 < 12 (𝑛 − 1)(𝑛 + 1) = 12 (𝑛 − 𝛾) (𝑛 − 𝛾 + 2). However, if 𝛾 ≥ 2, then Vizing [733] showed that the bound is achievable and constructed the following family ofgraphs 𝐻𝑛,𝛾 of order𝑛 and size 𝑚 with domination number 𝛾 ≥ 2 that satisfy 𝑚 = 12 (𝑛 − 𝛾) (𝑛 − 𝛾 + 2) . If 𝛾 = 2, then let 𝐻𝑛,2 be the graph obtained as follows. For 𝑛 even, let 𝐻𝑛,2 be obtained from a complete graph 𝐾𝑛 by removing the edges of a perfect matching. For 𝑛 ≥ 3 odd, let 𝐻𝑛,2 be obtained from a complete graph 𝐾𝑛 by deleting a maximum matching (of size (𝑛 − 1)/2) and an additional edge incident to the remaining vertex, which is not incident to an edge of the removed matching, that is, 𝐻𝑛,2 is obtained from 𝐾𝑛 by
removing from it the edges of a spanning subgraph isomorphic to 𝑃3 ∪ 𝑛−1 2 𝑃2 . Equivalently, for 𝑛 ≥ 2 the graph 𝐻 𝑛,2 is obtained from a complete graph 𝐾𝑛 by removing a minimum edge cover. The graphs 𝐻8,2 and 𝐻7,2 , for example, are shown in Figure 8.1(a) and (b), respectively.

(a) 𝐻8,2

(b) 𝐻7,2

Figure 8.1 The graphs 𝐻8,2 and 𝐻7,2 For all 𝑛 ≥ 2, the graph 𝐻𝑛,2 has size         𝑛 𝑛 𝑛(𝑛 − 2) (𝑛 − 𝛾) (𝑛 − 𝛾 + 2) 𝑚(𝐻𝑛,2 ) = − = = . 2 2 2 2 For 𝛾 > 2, let 𝐻𝑛,𝛾 = 𝐻𝑛−𝛾+2,2 ∪ (𝛾 − 2)𝐾1 , that is, 𝐻𝑛,𝛾 is obtained from the disjoint union of 𝐻𝑛−𝛾+2,2 and 𝛾 − 2 copies of 𝐾1 . The graphs 𝐻11,5 = 𝐻8,2 ∪ 3𝐾1 and 𝐻10,5 = 𝐻7,2 ∪ 3𝐾1 , for example, are shown in Figure 8.2(a) and (b), respectively.

Chapter 8. Bounds in Terms of Size

230

(a) 𝐻11,5

(b) 𝐻10,5

Figure 8.2 The graphs 𝐻11,5 and 𝐻10,5

For 𝛾 > 2, the graph 𝐻𝑛,𝛾 has order 𝑛, domination number 𝛾, and size     𝑚(𝐻𝑛,𝛾 ) = 𝑚(𝐻𝑛−𝛾+2,2 ) = 12 (𝑛 − 𝛾 + 2 − 2)(𝑛 − 𝛾 + 2) = 12 (𝑛 − 𝛾) (𝑛 − 𝛾 + 2) . As a corollary of Vizing’s Theorem, we have the following upper bound for the domination number of a graph in terms of its order and size. Theorem 8.3 ([663]) If 𝐺 is a graph of order 𝑛 and size 𝑚, then √ 𝛾(𝐺) ≤ 𝑛 + 1 − 1 + 2𝑚. Proof Let 𝐺 be a graph of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾. By Theorem 8.2, we have (𝑛 − 𝛾) (𝑛 − 𝛾 + 2) − 2𝑚 ≥ 0, or equivalently, (𝑛 − 𝛾) 2 + 2(𝑛 − 𝛾) √ √ − 2𝑚 ≥ 0. Thus, since 𝑛 − 𝛾 ≥ 0, we have 𝑛 − 𝛾 ≥ −1 + 1 + 2𝑚, or 𝛾 ≤ 𝑛 + 1 − 1 + 2𝑚. For 𝛾 ≥ 2, the family of graphs 𝐻𝑛,𝛾 constructed by Vizing are disconnected and all have maximum degree Δ = 𝑛 − 𝛾, showing that the upper bound of Theorem 8.2 is tight when Δ = 𝑛 − 𝛾. However, for smaller maximum degrees, in 1991 Sanchis [663] showed that the bound of Theorem 8.2 can be improved slightly. Theorem 8.4 ([663]) If 𝐺 is a graph of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾 ≥ 2 and Δ(𝐺) = Δ satisfying Δ ≤ 𝑛 − 𝛾 − 1, then 𝑚 ≤ 12 (𝑛 − 𝛾) (𝑛 − 𝛾 + 1). Proof For 1 ≤ Δ ≤ 𝑛 − 𝛾 − 1, let 𝑓 (Δ) = (𝑛 − 𝛾) (𝑛 − 𝛾 + 2) − Δ(𝑛 − 𝛾 − Δ). As shown in the proof of Theorem 8.2, we have 2𝑚 ≤ 𝑓 (Δ). The parabolic function 𝑓 (Δ) = Δ2 − (𝑛 − 𝛾)Δ + (𝑛 − 𝛾) (𝑛 − 𝛾 + 2) achieves its maximum value at one of its endpoints, namely Δ = 1 or Δ = 𝑛 − 𝛾 − 1. Since 𝑓 (1) = 𝑓 (𝑛 − 𝛾 − 1) = (𝑛 − 𝛾) (𝑛 − 𝛾 + 1) + 1, which is odd, and since 𝑚 ≤ 12 𝑓 (Δ) , it follows that 𝑚 ≤ 12 (𝑛 − 𝛾) (𝑛 − 𝛾 + 1). Vizing’s result in Theorem 8.2 was generalized in 1994 by Fulman [316] and others. For 𝛾 ≥ 2, the family of graphs 𝐻𝑛,𝛾 constructed by Vizing that achieve equality in the upper bound of Theorem 8.2 have two unusual properties. The first property is that they are disconnected, and the second property is that their edges are unevenly distributed in the sense that if 𝐺 is a graph in the family, then Δ(𝐺) = 0 while Δ(𝐺) = 𝑛 − 𝛾(𝐺). In 1991 Sanchis [663, 665] showed that if we restrict the

Section 8.2. Domination and Size

231

graph to being connected, then the bounds can be improved. However, the family of graphs that achieve the improved bound of Sanchis still have the second property that edges are unevenly distributed. In 1999 Rautenbach [649] showed that the square dependence on 𝑛 and 𝛾(𝐺) in Vizing’s result in Theorem 8.2 can be improved to a linear dependence on 𝑛, 𝛾(𝐺), and Δ(𝐺). For the remainder of this section, let Δ ≥ 3 be a given fixed integer, where Δ is not necessarily equal to the maximum degree Δ(𝐺). Theorem 8.5 ([649]) If 𝐺 is an isolate-free graph of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾 and Δ(𝐺) ≤ Δ, where Δ ≥ 3, then 𝑚 ≤ Δ𝑛 − (Δ + 1)𝛾.

(8.6)

Proof We proceed by induction on the order 𝑛 ≥ 2. If 𝑛 = 2, then 𝐺 = 𝐾2 and 𝑚 = 𝛾 = 1, and Δ𝑛 − (Δ + 1)𝛾 = 2Δ − (Δ + 1) = Δ − 1 ≥ 2 > 𝑚. This establishes the base case. Let 𝑛 ≥ 3 and suppose that the result is true for all isolate-free graphs of order less than 𝑛. Let 𝐺 be an isolate-free graph of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾, 𝛿(𝐺) = 𝛿, and Δ(𝐺) ≤ Δ, where Δ ≥ 3. If 𝐺 is disconnected with components 𝐺 1 , 𝐺 2 , . . . , 𝐺 𝑘 , where 𝑘 ≥ 2 and 𝐺 𝑖 has order 𝑛𝑖 , size 𝑚 𝑖 , and domination number 𝛾𝑖 for 𝑖 ∈ [𝑘], then applying the inductive hypothesis to each component, we have 𝑚=

𝑘 ∑︁ 𝑖=1

𝑚𝑖 ≤

𝑘 ∑︁


Δ𝑛𝑖 − (Δ + 1)𝛾𝑖 = Δ𝑛 − (Δ + 1)𝛾.

𝑖=1

Hence, since Inequality (8.6) is linear in 𝑚, 𝑛, and 𝛾, the result is immediate if 𝐺 is disconnected. Thus, we may assume that 𝐺 is connected. If Δ(𝐺) ≤ 2, then 𝐺 is a cycle or a path of order   at least 2. Thus, 𝑚 = 𝑛 if 𝐺 is a cycle or 𝑚 = 𝑛 − 1 is 𝐺 is a path. Further, 𝛾 = 31 𝑛 . Therefore,     𝑛  2𝑛   𝑛  Δ𝑛 − (Δ + 1)𝛾 = Δ𝑛 − (Δ + 1) 𝑛3 = Δ 2𝑛 3 − 3 ≥ 3 3 − 3 ≥ 𝑛 ≥ 𝑚, with strict inequality if 𝑛 ≠ 4 or Δ ≠ 3. Hence, if Δ(𝐺) ≤ 2, then Inequality (8.6) holds, and we have equality if and only if 𝐺 = 𝐶4 and Δ = 3. Hence, we may assume that Δ(𝐺) ≥ 3. Since Δ𝑛 − (Δ + 1)𝛾 is monotonically increasing in Δ, we can assume that Δ(𝐺) = Δ ≥ 3. (8.7) Let 𝑤 be a vertex of 𝐺 of minimum degree 𝛿 in 𝐺 and let 𝑣 be an arbitrary neighbor of 𝑤. Let 𝑉0 be the set of isolated vertices in 𝐺 − N[𝑣], let 𝑉1 be the set of vertices that are not isolated in 𝐺 − N[𝑣], and let 𝑉2 = N[𝑣] ∪ 𝑉0 . For 𝑖 ∈ [2], let 𝐺 𝑖 = 𝐺 [𝑉𝑖 ], where 𝐺 𝑖 has order 𝑛𝑖 , size 𝑚 𝑖 , and domination number 𝛾𝑖 . By construction, both 𝐺 1 and 𝐺 2 are isolate-free graphs. We note that Δ(𝐺 𝑖 ) ≤ Δ(𝐺) = Δ for 𝑖 ∈ [2]. Further, 𝑉 = 𝑉1 ∪𝑉2 , 𝑛 = 𝑛1 + 𝑛2 , and 𝛾 ≤ 𝛾1 + 𝛾2 . Let |𝑉0 | = 𝑛0 , and so 𝑛2 = deg𝐺 (𝑣) + 𝑛0 + 1. Thus, 𝑛1 = 𝑛 − 𝑛2 = 𝑛 − deg𝐺 (𝑣) − 𝑛0 − 1 and 𝛾1 ≥ 𝛾 − 𝛾2 . Applying the inductive hypothesis to 𝐺 1 , we have
𝑚 1 ≤ Δ𝑛1 − (Δ + 1)𝛾1 ≤ Δ 𝑛 − deg𝐺 (𝑣) − 𝑛0 − 1 − (Δ + 1) (𝛾 − 𝛾2 ). (8.8)

Chapter 8. Bounds in Terms of Size

232

Let 𝑚★ be the number of edges incident with a vertex in N(𝑣). By construction, every edge of 𝐺 belongs to 𝐺 1 or is incident with a vertex in N(𝑣), and so 𝑚 = 𝑚 1 +𝑚★. Thus, by Inequality (8.8),
𝑚 ≤ Δ𝑛 − (Δ + 1)𝛾 − Δ deg𝐺 (𝑣) + 𝑛0 + 1 + (Δ + 1)𝛾2 + 𝑚★ .

(8.9)

Therefore, if
𝑚★ + (Δ + 1)𝛾2 ≤ Δ deg𝐺 (𝑣) + 𝑛0 + 1 ,

(8.10)

then Inequality (8.6) follows from Inequality (8.9). Since the neighbor 𝑤 of the vertex 𝑣 has degree 𝛿 and each of the remaining deg𝐺 (𝑣) − 1 neighbors of 𝑣 has degree at most Δ,
(8.11) 𝑚★ ≤ 𝛿 + Δ deg𝐺 (𝑣) − 1 . Thus, by Inequalities (8.9) and (8.11), 𝑚 ≤ Δ𝑛 − (Δ + 1)𝛾 − 2Δ + 𝛿 − Δ𝑛0 + (Δ + 1)𝛾2 .

(8.12)

𝛿 + (Δ + 1)𝛾2 ≤ Δ(𝑛0 + 2),

(8.13)

Therefore, if

then Inequality (8.6) follows from Inequality (8.12). The set 𝑉0 ∪ {𝑣} is a dominating set of 𝐺 2 , implying that 𝛾2 ≤ 𝑛0 + 1 and therefore 𝛿 + (Δ + 1)𝛾2 ≤ 𝛿 + (Δ + 1) (𝑛0 + 1) = Δ(𝑛0 + 2) + (𝑛0 + 1 − Δ + 𝛿). Hence, if 𝑛0 ≤ Δ − 𝛿 − 1, then Inequality (8.13) holds, yielding Inequality (8.6). Therefore, we may assume that 𝑛0 ≥ Δ − 𝛿. Moreover, the set N(𝑣) is a dominating set of 𝐺 2 , implying that 𝛾2 ≤ deg𝐺 (𝑣) and therefore 𝛿 + (Δ + 1)𝛾2 ≤ Δ(𝑛0 + 2) + 𝛿 + (Δ + 1) deg𝐺 (𝑣) − 2Δ − Δ𝑛0 . Hence, if Δ𝑛0 ≥ (Δ+1) deg𝐺 (𝑣) + 𝛿 −2Δ, then Inequality (8.13) holds, once again yielding Inequality (8.6). Therefore, we may assume that Δ𝑛0 < (Δ + 1) deg𝐺 (𝑣) + 𝛿 − 2Δ. With these assumptions, and noting that deg𝐺 (𝑣) + 𝛿 − 2Δ ≤ 𝛿 − Δ ≤ 0, we have Δ − 𝛿 ≤ 𝑛0
1.15 × 1026 by proving the following result. Theorem 8.8 ([764]) For every Δ ≥ 3, there exists a bipartite Δ-regular graph 𝐺 of order 𝑛 with 𝛾(𝐺) = 𝛾 satisfying   0.05 ln(Δ) 𝛾> (8.16) 𝑛. Δ We note that if 𝐺 is a bipartite Δ-regular graph of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾 satisfying Inequality (8.16), then     Δ + 0.05 ln(Δ) Δ 0.05 ln(Δ) 1 𝑚 = 2 Δ𝑛 = 𝑛− 𝑛 2 2 Δ   Δ + 0.05 ln(Δ) Δ > 𝑛− 𝛾 2 2     Δ + 0.05 ln(Δ) Δ+5 > 𝑛− 𝛾. 2 2 Hence, as an immediate consequence of Theorem 8.8, we have the following result. Corollary 8.9 ([764]) For every Δ ≥ 3, there exists a bipartite Δ-regular graph 𝐺 of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾 satisfying     Δ + 0.05 ln(Δ) Δ+5 𝑚> 𝑛− 𝛾. 2 2 Corollary 8.9 disproves Conjecture 8.7 when 0.05 ln(Δ) > 3, which happens when Δ > 1.15 × 1026 . This gives rise to the following open problem. Problem 8.10 For each Δ ≥ 3, find the smallest value 𝑐 Δ such that for every isolate-free graph of order 𝑛 and size 𝑚 with 𝛾(𝐺) = 𝛾 and Δ(𝐺) ≤ Δ,     Δ + 𝑐Δ Δ + 𝑐Δ + 2 𝑚≤ 𝑛− 𝛾. 2 2

Chapter 8. Bounds in Terms of Size

238

Theorem 8.5, due to Rautenbach, implies that 𝑐 Δ ≤ Δ. Theorem 8.8, due to Yeo, implies that 𝑐 Δ > 0.05 ln(Δ). We state this formally as follows. Theorem 8.11 ([649, 764]) If Δ ≥ 3, then 0.05 ln(Δ) < 𝑐 Δ ≤ Δ. However, it remains an open problem to determine if 𝑐 Δ grows proportionally with ln(Δ) or some completely different function.

8.3

Total Domination and Size

In this section, we relate the size and the total domination number of a graph of given order 𝑛. To present our first such result, we shall need the following definition and lemma. For integers 𝑛 and 𝑘, where 𝑛 ≥ 𝑘 ≥ 1, let ( 𝑓 (𝑛, 𝑘) =

𝑛−𝑘+2
2 𝑛−𝑘+1
2

+ +

𝑘 2 𝑘 2

−1

if 𝑘 is even

1 2

if 𝑘 is odd.

+

The following properties of 𝑓 (𝑛, 𝑘) are readily determined from elementary arithmetic. Lemma 8.12 ([215]) The following properties of 𝑓 (𝑛, 𝑘) hold: (a) If 1 ≤ 𝑘 1 ≤ 𝑛1 and 1 ≤ 𝑘 2 ≤ 𝑛2 , then 𝑓 (𝑛1 + 𝑛2 , 𝑘 1 + 𝑘 2 ) ≥ 𝑓 (𝑛1 , 𝑘 1 ) + 𝑓 (𝑛2 , 𝑘 2 ), unless 𝑘 1 = 𝑛1 and 𝑘 1 is odd, or 𝑘 2 = 𝑛2 and 𝑘 2 is odd. (b) If 1 ≤ 𝑘 1 ≤ 𝑘 2 ≤ 𝑛, then 𝑓 (𝑛, 𝑘 2 ) ≤ 𝑓 (𝑛, 𝑘 1 ). (c) If 2 ≤ 𝑘 ≤ 𝑛 − 1, then 𝑓 (𝑛 − 1, 𝑘) < 𝑓 (𝑛, 𝑘) < 𝑓 (𝑛, 𝑘 − 1). (d) If 1 ≤ 𝑘 ≤ 𝑛 and 0 ≤ ℓ ≤ 𝑘 − 1, then 𝑓 (𝑛 − ℓ, 𝑘 − ℓ) ≤ 𝑓 (𝑛, 𝑘 − 1). (e) If 3 ≤ 𝑘 ≤ 𝑛 − 2 and 𝑘 is odd, then 𝑓 (𝑛 − 1, 𝑘) + 𝑛 − 𝑘 = 𝑓 (𝑛, 𝑘) = 𝑓 (𝑘 − 1, 𝑘 − 2) + (𝑛 − 𝑘) (𝑛 − 𝑘 + 1)/2. In 2004 Dankelmann et al. [215] established the following Vizing-like relation between the size and the total domination number of a graph of given order. Theorem 8.13 ([215]) If 𝐺 is an isolate-free graph of order 𝑛 and size 𝑚 with 𝛾t (𝐺) = 𝛾t ≥ 2, then 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). Proof We proceed by induction on the order 𝑛 of an isolate-free graph of size 𝑚 with 𝛾t (𝐺) = 𝛾t ≥ 2. The result is immediate if 𝑛 = 2 since in this case 𝛾t = 2 and 𝑚 = 1 = 𝑓 (2, 2) = 𝑓 (𝑛, 𝛾t ). This establishes the base case. Let 𝑛 ≥ 3 and assume that if 𝐺 ′ is an isolate-free graph having order 𝑛′ , size 𝑚 ′ , and total domination number 𝛾t′ , where 2 ≤ 𝑛′ < 𝑛, then 𝑚 ′ ≤ 𝑓 (𝑛′ , 𝛾t′ ). Let 𝐺 be an isolate-free graph of order 𝑛 and size 𝑚 with 𝛾t (𝐺) = 𝛾t and Δ(𝐺) = Δ. If
𝑛 is even, then 𝑛 ≥ 𝛾t ≥ 2, while if 𝑛 is odd, then 𝑛 > 𝛾t ≥ 2. If 𝛾t = 2, then 𝑚 ≤ 𝑛2 = 𝑓 (𝑛, 2) = 𝑓 (𝑛, 𝛾t ). Hence, we may assume that 𝛾t ≥ 3, for otherwise the desired result follows. We proceed with the following series of claims. Claim 8.13.1 If 𝐺 is disconnected, then 𝑚 ≤ 𝑓 (𝑛, 𝛾t ).

Section 8.3. Total Domination and Size

239

Proof Suppose that 𝐺 is disconnected. Let 𝐺 1 be a component of 𝐺 and let 𝐺 2 = 𝐺 −𝑉 (𝐺 1 ). Although 𝐺 2 may be disconnected, both 𝐺 1 and 𝐺 2 are isolate-free graphs. Let 𝐺 𝑖 have order 𝑛𝑖 and size 𝑚 𝑖 with total domination number 𝛾t𝑖 for 𝑖 ∈ [2]. Applying the inductive hypothesis to 𝐺 𝑖 , we have 𝑚 𝑖 ≤ 𝑓 (𝑛𝑖 , 𝛾t𝑖 ) for 𝑖 ∈ [2]. We note that a graph of odd order has total domination number strictly less than its order. Thus, if 𝑛𝑖 is odd, then 𝛾t𝑖 < 𝑛𝑖 for 𝑖 ∈ [2]. Hence, by Lemma 8.12(a), 𝑚 = 𝑚 1 + 𝑚 2 ≤ 𝑓 (𝑛1 , 𝛾t1 ) + 𝑓 (𝑛2 , 𝛾t2 ) ≤ 𝑓 (𝑛1 + 𝑛2 , 𝛾t1 + 𝛾t2 ) = 𝑓 (𝑛, 𝛾t ). By Claim 8.13.1, we may assume that 𝐺 is connected, for otherwise Theorem 8.13 holds. Recall that 𝛾t ≥ 3, implying that 𝑛 ≥ 4. By Theorem 6.40, we have 𝛾t ≤ 23 𝑛. In particular, we note that 𝛾t ≤ 𝑛 − 2. Let 𝑣 be a vertex of 𝐺 of maximum degree Δ and let 𝐴 = N(𝑣) = {𝑣 1 , 𝑣 2 , . . . , 𝑣 Δ } be the set of neighbors of 𝑣. Let 𝐵 = 𝑉 \ N[𝑣]. Since 𝐺 is connected and 𝛾t ≥ 3, 𝐵 must be nonempty. Claim 8.13.2 deg 𝐵 (𝑣 𝑖 ) ≤ 𝑛 − Δ − 𝛾t + 1 for all 𝑖 ∈ [Δ]. Proof Let 𝑖 ∈ [Δ], and let 𝐵𝑖 be the set of all vertices in 𝐵 that are not dominated by the vertex 𝑣 𝑖 , that is, 𝐵𝑖 = 𝐵 \ N 𝐵 (𝑣 𝑖 ). Let 𝐷 𝑖 = {𝑣, 𝑣 𝑖 } ∪ 𝐵𝑖 . If the graph 𝐺 [𝐷 𝑖 ] has no isolated vertices, then let 𝑆𝑖 = 𝐷 𝑖 . If the graph 𝐺 [𝐷 𝑖 ] contains an isolated vertex,
then the neighbors of such an isolated vertex belong to the set 𝐴 \ {𝑣 𝑖 } ∪ N 𝐵 (𝑣 𝑖 ). In this case, we replace each such isolated vertex in 𝐷 𝑖 with one of its neighbors, and let 𝑆𝑖 denote the resulting set. In both cases, the set 𝑆𝑖 is a TD-set of 𝐺 and 𝛾t ≤ |𝑆𝑖 | ≤ |𝐷 𝑖 | = 𝑛 − Δ + 1 − deg 𝐵 (𝑣 𝑖 ). Claim 8.13.3 Δ ≤ 𝑛 − 𝛾t . Proof By our earlier assumptions, 𝐺 is connected and 𝛾t ≥ 3, which implies that 𝐵 ≠ ∅ and at least one vertex in 𝐴 has a neighbor in 𝐵. Renaming vertices if necessary, we may assume that deg 𝐵 (𝑣 1 ) ≥ 1. By Claim 8.13.2, we have 𝛾t ≤ 𝑛 − Δ + 1 − deg 𝐵 (𝑣 1 ) ≤ 𝑛 − Δ.
Our next claim gives an upper bound on the size 𝑚 𝐺 [𝐵] of the graph 𝐺 [𝐵].
Claim 8.13.4 𝑚 𝐺 [𝐵] ≤ 𝑓 (𝑛 − Δ − 1, 𝛾t − 2). Proof Let 𝐼 𝐵 be the set of isolated vertices in 𝐺 [𝐵] and let |𝐼 𝐵 | = ℓ. If 𝐼 𝐵 is a proper subset of 𝐵, then let 𝐺 ′ = 𝐺 [𝐵\𝐼 𝐵 ]. Let 𝐺 ′ have order 𝑛′ , size 𝑚 ′ , and total domination number 𝛾t′ . Let 𝑆 ′ be a 𝛾t -set of 𝐺 ′ and so 𝛾t′ = |𝑆 ′ |. Applying the inductive hypothesis to 𝐺 ′ , we have 𝑚 ′ ≤ 𝑓 (𝑛′ , 𝛾t′ ). If ℓ = 0, then 𝑛′ = 𝑛 − Δ − 1 and the set 𝑆 ′ ∪ {𝑣, 𝑣 1 } is a TD-set of 𝐺, implying that 𝛾t ≤ |𝑆 ′ | + 2 = 𝛾t′ + 2. Therefore, 1 ≤ 𝛾t − 2 ≤ 𝛾t′ ≤ 𝑛′ . Thus, by Lemma 8.12(b), we have 𝑚 ′ ≤ 𝑓 (𝑛′ , 𝛾t′ ) ≤ 𝑓 (𝑛 − Δ − 1, 𝛾t − 2). Hence, we may assume that ℓ ≥ 1. Let 𝑢 ∈ 𝐼 𝐵 be an isolated vertex in 𝐺 [𝐵] having at least one neighbor in 𝐴. Renaming vertices if necessary, we may assume that 𝑢𝑣 1 is an edge. Let 𝑆 =
𝑆 ′ ∪ {𝑣, 𝑣 1 } ∪ 𝐼 𝐵 \ {𝑢} . If the graph 𝐺 [𝑆] has no isolated vertices, then let 𝐷 = 𝑆. If the graph 𝐺 [𝑆] has an isolated vertex, then such a vertex belongs to 𝐼 𝐵 and its neighbors belong to the set 𝐴 \ {𝑣 1 }. In this case, we replace each such isolated vertex

Chapter 8. Bounds in Terms of Size

240

in 𝑆 with one of its neighbors and let 𝐷 denote the resulting set. In both cases, the set 𝐷 is a TD-set of 𝐺 and 𝛾t ≤ |𝐷| ≤ |𝑆| = |𝑆 ′ | + 2 + (ℓ − 1) = 𝛾t′ + ℓ + 1 and so 𝛾t − ℓ − 1 ≤ 𝛾t′ . We note that 𝑛′ = 𝑛 − Δ − 1 − ℓ. Thus, by Lemma 8.12(b) and (d),
𝑚 𝐺 [𝐵] = 𝑚 ′ ≤ 𝑓 (𝑛′ , 𝛾t′ ) ≤ 𝑓 (𝑛 − Δ − 1 − ℓ, 𝛾t − 1 − ℓ) ≤ 𝑓 (𝑛 − Δ − 1, 𝛾t − 2), which yields the desired inequality, noting that 𝛾t − 2 ≥ 1 and the definition of 𝑓 (𝑛, 𝑘) also applies for 𝑘 = 1. By Claim 8.13.2, 2𝑚 = deg𝐺 (𝑣) +

Δ ∑︁

deg𝐺 (𝑣 𝑖 ) + | [ 𝐴, 𝐵] | + 2𝑚 𝐺 [𝐵]




𝑖=1

≤ Δ + Δ2 +

Δ ∑︁

deg 𝐵 (𝑣 𝑖 ) + 2𝑚 𝐺 [𝐵]




𝑖=1


≤ Δ + Δ + Δ(𝑛 − Δ − 𝛾t + 1) + 2𝑚 𝐺 [𝐵] , 2

or equivalently,
2𝑚 ≤ Δ(𝑛 − 𝛾t + 2) + 2𝑚 𝐺 [𝐵] .

(8.17)

By Claim 8.13.4 and Inequality (8.17), 2𝑚 ≤ Δ(𝑛 − 𝛾t + 2) + 2 𝑓 (𝑛 − Δ − 1, 𝛾t − 2).

(8.18)

By Claim 8.13.3 and our earlier observations, 2 ≤ Δ ≤ 𝑛 − 𝛾t . Claim 8.13.5 If 𝛾t is even, then 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). Proof Suppose that 𝛾t is even. The right-hand side of Inequality (8.18) is a parabola as a function of Δ. Since the second derivative of this function is positive, the function is maximized at an extremum, namely at Δ = 2 or Δ = 𝑛 − 𝛾t . The extremum is 2 𝑓 (𝑛, 𝛾t ), which occurs at Δ = 2, unless 𝛾t ≥ 𝑛 − 1. However, by our earlier observations, we have 𝛾t ≤ 𝑛 − 2, and so 𝛾t ≥ 𝑛 − 1 is not possible. Hence, by Inequality (8.18), we have 2𝑚 ≤ 2 𝑓 (𝑛, 𝛾t ), or equivalently, 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). By Claim 8.13.5, we may assume that 𝛾t is odd, for otherwise Theorem 8.13 holds. Claim 8.13.6 If Δ ≤ 𝑛 − 𝛾t − 1, then 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). Proof Suppose that Δ ≤ 𝑛−𝛾t −1. As in the proof of Claim 8.13.5, the right-hand side of Inequality (8.18) is maximized at an extremum, namely at Δ = 2 or Δ = 𝑛 − 𝛾t − 1. The extremum at these two values is the same value, namely 2 𝑓 (𝑛, 𝛾t ) + 2. Hence, by Inequality (8.18), we have 2𝑚 ≤ 2 𝑓 (𝑛, 𝛾t ) + 2, that is, 𝑚 ≤ 𝑓 (𝑛, 𝛾t ) + 1. Suppose that 𝑚 = 𝑓 (𝑛, 𝛾t ) + 1. With this assumption, we must have equality in Inequality (8.18), which in turn implies that we have equality in Claim 8.13.2, that is, deg 𝐵 (𝑣 𝑖 ) = 𝑛 − Δ − 𝛾t + 1 for all 𝑖 ∈ [Δ]. Thus, the proof of Claim 8.13.2 implies that the number of vertices needed to totally dominate the set 𝐵1 = 𝐵 \ N(𝑣 1 )

Section 8.3. Total Domination and Size

241

equals |𝐵1 |. Therefore, 𝐺 [𝐵1 ] is the disjoint union of copies of 𝐾2 . Further, every vertex in N
𝐵 (𝑣 1 ) has at most one neighbor in 𝐵1 . Hence, the graph 𝐺 [𝐵1 ] has size 𝑚 𝐺 [𝐵1 ] = 21 |𝐵1 | and | [𝐵1 , N 𝐵 (𝑣 1 )] | ≤ |N𝐺 (𝑣 1 )| = deg 𝐵 (𝑣 1 ). Therefore,  
deg 𝐵 (𝑣 1 ) 2𝑚 𝐺 [𝐵] ≤ |𝐵1 | + 2 deg 𝐵 (𝑣 1 ) + 2 2
= 𝛾t − 2 + deg 𝐵 (𝑣 1 ) deg 𝐵 (𝑣 1 ) + 1 = 𝛾t − 2 + (𝑛 − Δ − 𝛾t + 1) (𝑛 − Δ − 𝛾t + 2). By Inequality (8.17), 2𝑚 ≤ Δ(𝑛 − 𝛾t + 2) + 𝛾t − 2 + (𝑛 − Δ − 𝛾t + 1) (𝑛 − Δ − 𝛾t + 2).

(8.19)

Calculus arguments show that the right-hand side of Inequality (8.19) is maximized at an extremum, namely at Δ = 2 or Δ = 𝑛 − 𝛾t − 1. The extremum at these two values is the same value, namely 2 𝑓 (𝑛, 𝛾t ) + 1. Thus, 2𝑚 ≤ 2 𝑓 (𝑛, 𝛾t ) + 1, implying that 𝑚 ≤ 𝑓 (𝑛, 𝛾t ), a contradiction. Hence, our assumption that 𝑚 = 𝑓 (𝑛, 𝛾t ) + 1 is incorrect, implying that 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). By Claim 8.13.6, we may assume that Δ = 𝑛 − 𝛾t , for otherwise Theorem 8.13 holds. Claim 8.13.7 If there exists a pair of vertices 𝑥 and 𝑦 with N(𝑥) \ {𝑦} ⊆ N(𝑦) \ {𝑥} such that the graph 𝐺 − 𝑦 has no isolated vertices, then 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). Proof Let 𝑥 and 𝑦 be vertices of 𝐺 such that N(𝑥) \ {𝑦} ⊆ N(𝑦) \ {𝑥}. Let 𝐺 ′ = 𝐺 − 𝑦 and suppose, to the contrary, that 𝐺 ′ has no isolated vertices. Let 𝐺 ′ have order 𝑛′ , size 𝑚 ′ , and total domination number 𝛾t′ . Since every TD-set of 𝐺 ′ is a TD-set of 𝐺, 3 ≤ 𝛾t ≤ 𝛾t′ ≤ 𝑛′ ≤ 𝑛 − 1. By Lemma 8.12(b) and (c), we have 𝑚 ′ ≤ 𝑓 (𝑛′ , 𝛾t′ ) ≤ 𝑓 (𝑛′ , 𝛾t ) ≤ 𝑓 (𝑛 − 1, 𝛾t ). Recall that 𝛾t is odd. By Lemma 8.12(d), 𝑚 = 𝑚 ′ + deg𝐺 (𝑦) ≤ 𝑓 (𝑛 − 1, 𝛾t ) + Δ = 𝑓 (𝑛 − 1, 𝛾t ) + 𝑛 − 𝛾t = 𝑓 (𝑛, 𝛾t ), which yields the desired inequality. By Claim 8.13.7, if 𝑥 and 𝑦 are distinct vertices of 𝐺 with N(𝑥) \ {𝑦} ⊆ N(𝑦) \ {𝑥}, then we may assume that the graph 𝐺 − 𝑦 contains isolated vertices, for otherwise the desired result holds. Claim 8.13.8 If the graph 𝐺 − 𝑣 has an isolated vertex, then 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). Proof Suppose that 𝐺 − 𝑣 has an isolated vertex. Renaming vertices in 𝐴 if necessary, we may assume that 𝑣 1 is isolated in 𝐺 − 𝑣 and deg𝐺 (𝑣 1 ) = 1. Since we may
assume by Claim 8.13.6 that Δ = 𝑛 − 𝛾t , by Claim 8.13.4, we have 𝑚 𝐺 [𝐵] ≤ 𝑓 (𝑛 − Δ − 1, 𝛾t − 2) = 𝑓 (𝛾t − 1, 𝛾t − 2). Returning to our derivation of Inequality (8.17), we now have by Lemma 8.12(e) that

Chapter 8. Bounds in Terms of Size

242 2𝑚 = deg𝐺 (𝑣) + deg𝐺 (𝑣 1 ) +

Δ ∑︁

deg𝐺 (𝑣 𝑖 ) + | [ 𝐴, 𝐵] | + 2𝑚 𝐺 [𝐵]




𝑖=2 Δ ∑︁

deg 𝐵 (𝑣 𝑖 ) + 2𝑚 𝐺 [𝐵]




≤ Δ + 1 + (Δ − 1) (𝑛 − Δ − 𝛾t + 1) + 2𝑚 𝐺 [𝐵]
≤ (𝑛 − 𝛾t ) (𝑛 − 𝛾t + 1) + 1 + 2𝑚 𝐺 [𝐵] ≤ (𝑛 − 𝛾t ) (𝑛 − 𝛾t + 1) + 1 + 2 𝑓 (𝛾t − 1, 𝛾t − 2) = 2 𝑓 (𝑛, 𝛾t ) + 1.




≤ Δ + 1 + (Δ − 1)Δ +

𝑖=2 2

Since 𝑓 (𝑛, 𝛾t ) is an integer, it follows that 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). By Claim 8.13.8, we may assume that the graph 𝐺 − 𝑣 has no isolated vertices. Claim 8.13.9 The following hold: (a) Each vertex in 𝐴 has exactly one neighbor in 𝐵. (b) | [ 𝐴, 𝐵] | = Δ. Proof By our earlier assumptions, the graph 𝐺 − 𝑣 has no isolated vertices. If there is a vertex 𝑣 𝑖 ∈ 𝐴 such that N(𝑣 𝑖 )\{𝑣} ⊆ 𝐴, then by Claim 8.13.7, we have 𝑚 ≤ 𝑓 (𝑛, 𝛾t ). Hence, we may assume that no neighbor of 𝑣 has all its other neighbors in 𝐴. With this assumption, every neighbor of 𝑣 has a neighbor in 𝐵, and so deg 𝐵 (𝑣 𝑖 ) ≥ 1 for all 𝑖 ∈ [Δ]. By Claim 8.13.2 and our earlier assumption that Δ = 𝑛 − 𝛾t , we have deg 𝐵 (𝑣 𝑖 ) ≤ 𝑛 − Δ − 𝛾t + 1 = 1 for all 𝑖 ∈ [Δ]. Consequently, deg 𝐵 (𝑣 𝑖 ) = 1 for all 𝑖 ∈ [Δ]. This proves part (a). Part (b) is an immediate consequence of part (a). Let 𝐺 ′ = 𝐺 [𝐵] and let 𝐺 ′ have order 𝑛′ , size 𝑚 ′ , and total domination number 𝛾t′ . We note that 𝑛′ = |𝐵| = 𝑛 − Δ − 1 = 𝛾t − 1. By our earlier assumption, 𝛾t is odd and so 𝑛′ is even. Claim 8.13.10 The following hold: (a) Every component of 𝐺 ′ is a 𝐾2 . (b) 𝑚 ′ = 12 𝑛′ = 12 (𝛾t − 1). (c) Every vertex in 𝐵 has at least one neighbor in 𝐴. Proof (a) Since the graph 𝐺 − 𝑣 is isolate-free, no vertex in 𝐵 has all of its neighbors in 𝐴. Thus, every vertex in 𝐵 has at least one neighbor in 𝐵, and so the graph 𝐺 [𝐵] is isolate-free. If 𝛾t′ ≤ 𝑛′ − 2, then since every 𝛾t -set of 𝐺 ′ can be extended to a TD-set of 𝐺 by adding the vertices 𝑣 and 𝑣 1 to it, this would imply that 𝑛′ + 1 = 𝛾t ≤ 𝛾t′ + 2 ≤ 𝑛′ , a contradiction. Hence, 𝛾t′ ≥ 𝑛′ − 1, implying that every component of 𝐺 [𝐵] is a 𝐾2 , except possibly for one component which is either a 𝑃3 or a 𝐾3 . However, 𝑛′ is even, implying that every component of 𝐺 [𝐵] is a 𝐾2 . This proves (a). (b) By part (a), we have 𝑚 ′ = 12 𝑛′ = 12 (𝛾t − 1). (c) Suppose, to the contrary, that there is a vertex 𝑥 ∈ 𝐵 with no neighbor in 𝐴. By part (a), every component of 𝐺 [𝐵] is a 𝐾2 . Let 𝑥 ′ be the neighbor of 𝑥 in 𝐵. Since 𝐺 is connected, the vertex 𝑥 ′ has a neighbor 𝑦 ∈ 𝐴. Thus, 𝑦 = 𝑣 𝑖 for some 𝑖 ∈ [Δ].

Section 8.3. Total Domination and Size

243

We note that N(𝑥) = {𝑥 ′ } ⊂ N(𝑦). However, the graph 𝐺 − 𝑦 does not contain an isolated vertex, contradicting our earlier assumptions. Hence, every vertex in 𝐵 has a neighbor in 𝐴.

Claim 8.13.11 𝑚 𝐺 [ 𝐴] ≤ Δ2 − (Δ − 1). Proof By Claim 8.13.9(a), every vertex in 𝐴 has exactly one neighbor in 𝐵, and by Claim 8.13.10(c), every vertex in 𝐵 has at least one neighbor in 𝐴. Let 𝐵 = {𝑤 1 , 𝑤 2 , . . . , 𝑤 𝑛′ }, where we note that 𝑛′ ≥ 2. Let 𝑁𝑖 = N 𝐴 (𝑤 𝑖 ) for 𝑖 ∈ [𝑛′ ]. By our earlier observations, {𝑁1 , 𝑁2 , . . . , 𝑁 𝑛′ } is a partition of the set 𝐴. Let 𝐷 be a set of vertices in 𝐴 that contains exactly one vertex from each of the sets 𝑁𝑖 , where 𝑖 ∈ [𝑛′ ]. We note that |𝐷| = |𝐵| = 𝑛′ = 𝛾t − 1, implying that the set 𝐷 is not a TD-set of 𝐺. By construction of the set 𝐷, every vertex in 𝐵 ∪ {𝑣} is totally dominated by the set 𝐷. Hence, there exists a vertex 𝑧 ∈ 𝐴 that is not totally dominated by 𝐷, where possibly, 𝑧 ∈ 𝐷. Let 𝑦 𝑖 and 𝑦 𝑗 be an arbitrary pair of vertices in 𝑁𝑖 and 𝑁 𝑗 , respectively, where 1 ≤ 𝑖 < 𝑗 ≤ 𝑛′ . Consider the subgraph 𝐺 [ 𝐴] of the complement 𝐺 of 𝐺 induced by the set 𝐴. If 𝑦 𝑖 and 𝑦 𝑗 are not adjacent in 𝐺, then 𝑦 𝑖 and 𝑦 𝑗 are adjacent in the complement 𝐺 [ 𝐴]. If 𝑦 𝑖 and 𝑦 𝑗 are adjacent in 𝐺, then 𝑧 is distinct from 𝑦 𝑖 and 𝑦 𝑗 , implying that 𝑦 𝑖 𝑧 𝑦 𝑗 is a path in the complement 𝐺 [ 𝐴]. In both cases, 𝑦 𝑖 and 𝑦 𝑗 are connected in 𝐺 [ 𝐴]. Since 𝑦 𝑖 and 𝑦 𝑗 are arbitrary vertices in 𝑁𝑖 and 𝑁 𝑗 , respectively, where 1 ≤ 𝑖 < 𝑗 ≤ 𝑛′ , and since 𝑛′ ≥ 2, this implies that the subgraph 𝐺 [ 𝐴] is connected. Hence, 𝐺 [ 𝐴] contains at least | 𝐴| − 1 =
Δ −
1 edges, and therefore 𝐺 [ 𝐴] has at least Δ − 1 missing edges, that is, 𝑚 𝐺 [ 𝐴] ≤ Δ2 − (Δ − 1).
By Claim 8.13.10(b), we have 𝑚 𝐺 [𝐵] = 𝑚 ′ = 12 (𝛾t − 1). By our earlier assumption that Δ = 𝑛 − 𝛾t and by Claims 8.13.4, 8.13.9(b), and 8.13.11,

𝑚 = deg𝐺 (𝑣) + 𝑚 𝐺 [ 𝐴] + | [ 𝐴, 𝐵] | + 𝑚 𝐺 [𝐵]
≤ 𝛿 + 2𝛿 − (𝛿 − 1) + 𝛿 + 21 (𝛾t − 1) 𝛿(𝛿 + 1) 𝛾t 1 = + + 2 2 2 (𝑛 − 𝛾t ) (𝑛 − 𝛾t + 1) 𝛾t 1 = + + 2 2 2 = 𝑓 (𝑛, 𝛾t ). The bound in Theorem 8.13 is tight, as may be seen by considering the graph 𝐺 (𝑛, 𝛾t ) defined as follows. For 𝛾t ≥ 2 even and 𝑛 ≥ 𝛾t , let 𝐺 (𝑛, 𝛾t ) be the disjoint union of 𝐾𝑛−𝛾t +2 and 12 (𝛾t − 2) copies of 𝐾2 . For 𝛾t ≥ 3 odd and 𝑛 ≥ 𝛾t + 1, let 𝐺 (𝑛, 𝛾t ) be the graph obtained from 𝐺 (𝑛 − 2, 𝛾t − 1) by subdividing one edge of the component isomorphic to 𝐾𝑛−𝛾t +1 twice. The graphs 𝐺 (11, 8) and 𝐺 (13, 9), for example, are illustrated in Figure 8.4(a) and (b), respectively. In the case where 𝛾t ≥ 2 is even, the graph 𝐺 (𝑛, 𝛾t ) has order 𝑛, total domination number 𝛾t , and size   𝑛 − 𝛾t + 2 𝛾t 𝑚= + − 1 = 𝑓 (𝑛, 𝛾t ). 2 2

Chapter 8. Bounds in Terms of Size

244

(a) 𝐺 (11, 8)

(b) 𝐺 (13, 9)

Figure 8.4 The graphs 𝐺 (11, 8) and 𝐺 (13, 9)

In the case where 𝛾t ≥ 3 is even, the graph 𝐺 (𝑛, 𝛾t ) has order 𝑛, total domination number 𝛾t , and size     𝑛 − 𝛾t + 1 𝛾t − 3 𝑛 − 𝛾t + 1 𝛾t 1 𝑚= +2+ = + + = 𝑓 (𝑛, 𝛾t ). 2 2 2 2 2 Thus, in both cases, the graph 𝐺 (𝑛, 𝛾t ) has order 𝑛, total domination number 𝛾t , and size 𝑚 = 𝑓 (𝑛, 𝛾t ). The bound in Theorem 8.13 is therefore tight. For 𝛾t ≥ 2, the extremal graphs 𝐺 (𝑛, 𝛾t ) that achieve equality in the bound have two unusual properties, which is similar to the situation that arose with the extremal graphs 𝐻𝑛,𝛾 that achieve equality in the upper bound of Theorem 8.2. The first property is that the extremal graphs 𝐺 (𝑛, 𝛾t ) are disconnected, and the second property is that their edges are very unevenly distributed in the sense that 𝛿(𝐺) and Δ(𝐺) differ greatly. Indeed, for 𝛾t ≥ 3, the graph 𝐺 = 𝐺 (𝑛, 𝛾t ) has minimum degree 𝛿(𝐺) = 1 and maximum degree Δ(𝐺) = 𝑛 − 𝛾t + 1. In 2004 Sanchis [666] improved the bound of Theorem 8.13 slightly in the case when the graph 𝐺 is connected and 𝛾t (𝐺) ≥
5. To state her result, for 𝑛 ≥ 1, let 𝐹𝑛 = 𝑛2 𝐾2 if 𝑛 is even and let 𝐹𝑛 = 𝐾1 ∪ 𝑛−1 2 𝐾2 if 𝑛 is odd. Theorem 8.14 ([666]) If 𝐺 is a connected graph of order 𝑛 and size 𝑚 with 𝛾t (𝐺) = 𝛾t , then     𝑛 − 𝛾t + 1 𝛾t 𝑚≤ + . 2 2 Further, if 𝐺 achieves equality in this bound, then it has one of the following forms: (a) The graph 𝐺 is obtained from 𝐾𝑛−𝛾t ∪ 𝐹𝛾t by adding edges between the clique and the graph 𝐹𝛾t in such a way that each vertex in the clique is adjacent to exactly one vertex in 𝐹𝛾t and each component of 𝐹𝛾t has at least one vertex adjacent to a vertex in the clique. (b) For 𝛾t = 5 and 𝑛 ≥ 9, the graph 𝐺 is obtained from 𝐾𝑛−7 ∪ 𝑃3 ∪ 𝑃4 by joining every vertex in the clique to both ends of the 𝑃4 and to exactly one end of the 𝑃3 in such a way that each end of the 𝑃3 is adjacent to at least one vertex in the clique. (c) For 𝛾t = 6 and 𝑛 ≥ 9, 𝐺 is obtained from 𝐾𝑛−6 ∪ 𝐹6 by letting 𝑆 be a maximum independent set in 𝐹6 and joining every vertex in the clique to exactly two vertices of 𝑆 in such a way that each vertex in 𝑆 is adjacent to at least one vertex in the clique.

Section 8.3. Total Domination and Size

245

Examples of graphs constructed in the statement of Theorem 8.14(a), (b), and (c) are shown in Figure 8.5(a), (b), and (c), respectively, where some of the edges of the complete subgraphs are omitted for clarity. 𝐾8

𝐾6

𝐾5

(a)

(b)

(c)

Figure 8.5 Examples of graphs in the statement of Theorem 8.14 The graphs that achieve equality in the improved bound of Sanchis in Theorem 8.14 are also very unevenly distributed in the sense that the extremal graphs 𝐺 have small minimum degree and large maximum degree, namely Δ(𝐺) = 𝑛 − 𝛾t 𝐺. In 2005 Henning [458] and in 2007 Shan et al. [672] established a linear Vizinglike theorem relating the size of a graph and its order, total domination number, and maximum degree. Their results showed that the square dependence on 𝑛 and 𝛾t in Theorem 8.13 and Theorem 8.14 can be improved to a linear dependence on 𝑛, 𝛾t (𝐺), and the maximum degree Δ(𝐺). For the remainder of this section, let Δ ≥ 3 be a given fixed integer, where Δ is not necessarily equal to the maximum degree Δ(𝐺). Theorem 8.15 ([458, 672]) Let 𝐺 be a graph of order 𝑛 and size 𝑚 with 𝛾t (𝐺) = 𝛾t and Δ(𝐺) ≤ Δ, where Δ ≥ 3. If every component of 𝐺 has order at least 3, then 𝑚 ≤ Δ(𝑛 − 𝛾t ).

(8.20)

Proof We proceed by induction on the order 𝑛 ≥ 3. If 𝑛 = 3, then 𝐺 = 𝑃3 or 𝐺 = 𝐾3 . In both cases, 𝛾t = Δ(𝐺) = 2, and so Δ(𝑛 − 𝛾t ) = Δ ≥ 3 ≥ 𝑚. This establishes the base case. Let 𝑛 ≥ 4 and suppose that the result is true for all graphs in which every component has order at least 3. Let 𝐺 be a graph, each component of which has order at least 3, of order 𝑛 and size 𝑚 with Δ(𝐺) ≤ Δ, where Δ ≥ 3. Since Inequality (8.20) is linear in 𝑚, 𝑛, and 𝛾t , the result is immediate if 𝐺 is disconnected. Hence, we may assume that 𝐺 is connected. IfΔ(𝐺)  ≤  2,then 𝐺 is a cycle or a path of order at least 4 and 𝑚 ≤ 𝑛. In this case, 𝛾t = 12 𝑛 + 14 𝑛 − 14 𝑛 ≤ 23 𝑛, and so Δ(𝑛 − 𝛾t ) ≥ 3(𝑛 − 𝛾t ) ≥ 𝑛 ≥ 𝑚. Let 𝐺 have minimum degree 𝛿(𝐺) = 𝛿. If 𝛿 ≥ 3, then by Theorem 6.52, we have 𝛾t ≤ 12 𝑛 and so Δ(𝑛 − 𝛾t ) ≥ 3(𝑛 − 𝛾t ) ≥ 32 𝑛 ≥ 𝑚. Hence, we may assume that 𝛿 ∈ [2] and Δ(𝐺) ≥ 3, for otherwise Inequality (8.20) holds as desired. Since Δ(𝑛 − 𝛾t ) is monotonically increasing in Δ, we can assume that Δ(𝐺) = Δ ≥ 3.

(8.21)

Let 𝑢 be a vertex of 𝐺 of minimum degree 𝛿 in 𝐺 and let 𝑣 be an arbitrary neighbor of 𝑢. Let 𝑉1 be the set of isolated vertices in 𝐺 − N[𝑣], and let 𝑉2 be the set of vertices

Chapter 8. Bounds in Terms of Size

246

in 𝐺 − N[𝑣]. Further, let 𝑛𝑖 = |𝑉𝑖 | for 𝑖 ∈ [2] and let that belong to 𝑃2 -components
𝑉 ′ = 𝑉 \ N[𝑣] ∪ 𝑉1 ∪ 𝑉2 . If 𝑉 ′ ≠ ∅, then let 𝐺 ′ = 𝐺 [𝑉 ′ ] and let 𝐺 ′ have order 𝑛′ , size 𝑚 ′ , and total domination number 𝛾t′ . Let 𝐻 = 𝐺 − 𝑉 (𝐺 ′ ) = 𝐺 [N[𝑣] ∪ 𝑉1 ∪ 𝑉2 ]. We note that 𝑛 = 𝑛(𝐻) + 𝑛′ = deg𝐺 (𝑣) + 1 + 𝑛1 + 𝑛2 + 𝑛′ , 𝛾t ≤ 𝛾t (𝐻) + 𝛾t′ , and each of 𝐺 ′ and 𝐻 has maximum degree at most Δ. Applying the inductive hypothesis to 𝐺 ′ , we have 𝑚 ′ ≤ Δ(𝑛′ − 𝛾t′ ). Claim 8.15.1 If 𝑛1 + 𝑛2 ≥ 1, then the following hold: (a) 𝛾t (𝐻) ≤ 𝑛1 + 𝑛2 + 1. (b) If a vertex in N(𝑣) has two or more neighbors in 𝑉1 ∪ 𝑉2 , then 𝛾t (𝐻) ≤ 𝑛1 + 𝑛2 . (c) 𝛾t (𝐻) ≤ deg𝐺 (𝑣) + 21 𝑛2 + 1. (d) If 𝑛1 = 0, then 𝛾t (𝐻) ≤ deg𝐺 (𝑣) + 12 𝑛2 . Proof (a) By supposition, 𝐺 is connected and has order 𝑛 ≥ 4. Thus, each vertex in 𝑉1 has at least one neighbor in N(𝑣), while each 𝑃2 -component of 𝐺 [𝑉2 ] contains a vertex that has at least one neighbor in N(𝑣). Let 𝑈2 ⊂ 𝑉2 consist of one vertex from every 𝑃2 -component of 𝐺 [𝑉2 ] that has a neighbor in N(𝑣). Let 𝑆 be a minimum set in N(𝑣) that dominates the set 𝑉1 ∪ 𝑈2 . We note that |𝑆| ≤ |𝑉1 | + |𝑈2 |. Since 𝑆 ∪ 𝑈2 ∪ {𝑣} is a TD-set of 𝐻, 𝛾t (𝐻) ≤ |𝑆| + |𝑈2 | + 1 ≤ |𝑉1 | + 2|𝑈2 | + 1 = 𝑛1 + 𝑛2 + 1. (b) Suppose that a neighbor 𝑤 of 𝑣 has two or more neighbors in 𝑉1 ∪ 𝑉2 . We show that 𝛾t (𝐻) ≤ 𝑛1 + 𝑛2 . If 𝑤 is adjacent to at most one vertex from each 𝑃2 -component of 𝐺 [𝑉2 ], then we can choose the sets 𝑆 and 𝑈2 so that 𝑤 ∈ 𝑆 and |𝑆| ≤ |𝑉1 | + |𝑈2 | − 1, implying that 𝛾t (𝐻) ≤ 𝑛1 + 𝑛2 . Hence, we may assume that 𝑤 is adjacent to both vertices from the same 𝑃2 -component of 𝐺 [𝑉2 ]. If 𝑥 is the
vertex in such a 𝑃2 -component that belongs to the set 𝑈2 , then the set 𝑆 ∪ 𝑈2 \ {𝑥} ∪ {𝑣} is a TD-set of 𝐻. Therefore, 𝛾t (𝐻) ≤ |𝑆| + |𝑈2 | ≤ 𝑛1 + 𝑛2 . (c) Since N(𝑣) ∪ 𝑈2 ∪ {𝑣} is a TD-set of 𝐻, 𝛾t (𝐻) ≤ deg𝐺 (𝑣) + |𝑈2 | + 1 = deg𝐺 (𝑣) + 12 𝑛2 + 1. (d) Suppose that 𝑛1 = 0. As before, let 𝑆 be a minimum set in N(𝑣) that dominates the set 𝑈2 . If 𝑆 ≠ N(𝑣), then |𝑆| ≤ deg𝐺 (𝑣) − 1 and the set 𝑆 ∪ 𝑈2 ∪ {𝑣} is a TD-set of 𝐻, implying that 𝛾t (𝐻) ≤ |𝑆| + |𝑈2 | + 1 ≤ deg𝐺 (𝑣) + 12 𝑛2 . If 𝑆 = N(𝑣), then every vertex in N(𝑣) uniquely dominates a vertex in 𝑈2 that is not dominated by any other vertex in N(𝑣), implying that the set N(𝑣) ∪ 𝑈2 is a TD-set of 𝐻. Therefore, 𝛾t (𝐻) ≤ deg𝐺 (𝑣) + 12 𝑛2 . Claim 8.15.2 If 𝑛1 + 𝑛2 ≥ 1 and 𝛿 + 12 𝑛2 + Δ 𝛾t (𝐻) ≤ Δ(𝑛1 + 𝑛2 + 2),

(8.22)

then Inequality (8.20) holds. Proof Suppose that 𝑛1 + 𝑛2 ≥ 1 and that Inequality (8.22) holds. Let 𝑚 1 be the number of edges in 𝐺 incident with vertices in N(𝑣). Since the vertex 𝑢 ∈ N(𝑣) has
degree 𝛿 in 𝐺, 𝑚 1 ≤ 𝛿 + Δ deg𝐺 (𝑣) − 1 . By construction, every edge of 𝐺 is incident with a vertex in N(𝑣) or belongs to 𝐺 ′ or to a 𝑃2 -component in 𝐺 [𝑉2 ]. Recall that 𝑛 = deg𝐺 (𝑣) + 1 + 𝑛1 + 𝑛2 + 𝑛′ and 𝛾t ≤ 𝛾t (𝐻) + 𝛾t′ . These observations, together with our supposition that Inequality (8.22) holds, imply that the number of edges 𝑚 of 𝐺 satisfies

Section 8.3. Total Domination and Size

247


𝑚 = 𝑚 1 + 𝑚 𝐺 [𝑉2 ] + 𝑚 ′ ≤ 𝑚 1 + 21 𝑛2 + Δ(𝑛′ − 𝛾𝑡′ )

≤ 𝛿 + Δ deg𝐺 (𝑣) − 1 + 12 𝑛2 + Δ 𝑛 − deg𝐺 (𝑣) − 1 − 𝑛1 − 𝑛2 − 𝛾t + 𝛾t (𝐻) ≤ Δ(𝑛 − 𝛾𝑡 ) + 𝛿 + 12 𝑛2 + Δ𝛾t (𝐻) − Δ(𝑛1 + 𝑛2 + 2) ≤ Δ(𝑛 − 𝛾𝑡 ), that is, Inequality (8.20) holds. By Claim 8.15.2, we may assume that if 𝑛1 + 𝑛2 ≥ 1, then Inequality (8.22) does not hold, for otherwise Inequality (8.20) follows. With this assumption, we have the following range of possible values for 𝑛2 . Claim 8.15.3 If 𝑛1 + 𝑛2 ≥ 1, then 2(Δ − 𝛿 + 1) ≤ 𝑛2 < 2 deg𝐺 (𝑣). Proof Suppose that 𝑛1 + 𝑛2 ≥ 1. By Claim 8.15.1(a), we have 𝛾t (𝐻) ≤ 𝑛1 + 𝑛2 + 1. If 𝑛2 ≤ 2(Δ − 𝛿), then 𝛿 + 12 𝑛2 + Δ 𝛾t (𝐻) ≤ 𝛿 + (Δ − 𝛿) + Δ(𝑛1 + 𝑛2 + 1) = Δ(𝑛1 + 𝑛2 + 2), that is, Inequality (8.22) holds, a contradiction. Hence, since 𝑛2 is even, we have 𝑛2 ≥ 2(Δ − 𝛿 + 1). This establishes the lower bound on 𝑛2 . We prove next the upper bound on 𝑛2 . Assume that 𝑛1 ≥ 1. By Claim 8.15.1(c), we have 𝛾t (𝐻) ≤ deg𝐺 (𝑣) + 12 𝑛2 + 1. Thus, ⇑ ⇕ ⇕

𝛿 + 12 𝑛2 + Δ 𝛾t (𝐻) ≤ Δ(𝑛1 + 𝑛2 + 2)
𝛿 + 12 𝑛2 + Δ deg𝐺 (𝑣) + 12 𝑛2 + 1 ≤ Δ(𝑛1 + 𝑛2 + 2)   Δ+1 − Δ 𝑛2 ≤ Δ(𝑛1 + 1) − 𝛿 − Δ deg𝐺 (𝑣) 2
2 Δ deg𝐺 (𝑣) + 𝛿 − Δ − Δ𝑛1 𝑛2 ≥ . Δ−1


Thus, if 𝑛2 ≥ 2 Δ deg𝐺 (𝑣) + 𝛿 − Δ − Δ𝑛1 /(Δ − 1), then Inequality (8.22) holds, a contradiction. Hence, by our supposition that 𝑛1 ≥ 1, we have
2 Δ deg𝐺 (𝑣) + 𝛿 − Δ − Δ𝑛1 Δ−1
2 = 2 deg𝐺 (𝑣) + deg𝐺 (𝑣) + 𝛿 − Δ − Δ𝑛1 Δ−1
2 ≤ 2 deg𝐺 (𝑣) + Δ+𝛿−Δ−Δ Δ−1 ≤ 2 deg𝐺 (𝑣).

𝑛2
(8.24) 𝑛. Δ As a consequence of Theorem 8.21, we have the following result.

Section 8.4. Independent Domination and Size

255

Corollary 8.22 ([764]) For every Δ ≥ 3, 𝑟Δ >

0.1 ln(Δ) 1−

(8.25)

.

0.1 ln(Δ) Δ

Proof Suppose, to the contrary, that Inequality (8.25) is not true. Let 𝐺 be a bipartite Δ-regular graph 𝐺 with order 𝑛 and size 𝑚 satisfying Inequality (8.24). Thus,   0.1 ln(Δ) 0.1 ln(Δ) 𝑟Δ ≤ and 𝛾t (𝐺) > 𝑛. Δ 1 − 0.1 ln(Δ) Δ By the definition of 𝑟 Δ , 𝛾t ≤ 𝑛 −

2𝑚 Δ𝑛 ≤ 𝑛− Δ + 𝑟Δ Δ + 0.10.1ln(Δ) ln(Δ) 1−

= 1−

Δ

Δ Δ+

!

Δ0.1 ln(Δ) Δ−0.1 ln(Δ)

𝑛

  Δ − 0.1 ln(Δ) = 1− 𝑛 Δ   0.1 ln(Δ) = 𝑛, Δ a contradiction. Therefore, Inequality (8.25) holds. As an immediate consequence of Corollary 8.22, we have the slightly weaker, but simpler, bound on 𝑟 Δ . Corollary 8.23 ([764]) For every Δ ≥ 3, 𝑟 Δ > 0.1 ln(Δ). Corollary 8.23 disproves Conjecture 8.20 when 0.1 ln(Δ) > 3, which happens when Δ > 1.07 × 1013 . We summarize the results in this section as follows.  √ Theorem 8.24 ([458, 764]) For all Δ ≥ 3, 0.1 ln(Δ) < 𝑟 Δ ≤ max 3, 2 Δ . √It remains an open problem to determine if 𝑟 Δ grows proportionally with ln(Δ) or Δ, or some completely different function.

8.4

Independent Domination and Size

In this section, we relate the size and the independent domination number of a graph of given order. Recall that by Observation 6.80 in Section 6.4.1, if 𝐺 is a graph of order 𝑛 with maximum degree Δ, then we have the trivial bound 𝑖(𝐺) ≤ 𝑛 − Δ. Hence, if 𝐺 has size 𝑚, then 𝑚 ≤ 12 Δ𝑛, implying that 𝑖(𝐺) ≤ 𝑛 − 2𝑚 𝑛 , or equivalently,

𝑛
1 1 𝑚 ≤ 2 𝑛 𝑛 − 𝑖(𝐺) = 2 − 2 𝑛 𝑖(𝐺) − 1 . We state this formally as follows.

256

Chapter 8. Bounds in Terms of Size

Theorem 8.25 If 𝐺 is a graph of order 𝑛 and size 𝑚 with 𝑖(𝐺) = 𝑘, then
𝑚 ≤ 𝑛2 − 21 𝑛(𝑘 − 1). If 𝐺 is a complete multipartite graph of order 𝑛 and size 𝑚 with 𝑘 vertices in each partite set, then 𝐺 has independent domination number 𝑘 and size 𝑚 = 12 𝑛(𝑛 − 𝑘) = 𝑛
1 2 − 2 𝑛(𝑘 − 1). Hence, the trivial upper bound given in Theorem 8.25 is tight if 𝑛 is a multiple of 𝑘. In 2004 Dankelmann et al. [215] extended the result in Theorem 8.25 to handle the case when 𝑛 is not a multiple of 𝑘. Their key result is the following lemma. Lemma 8.26 ([215]) If 𝐺 is a graph of order 𝑛 and size 𝑚 such that every vertex is in a 𝑘-clique, then 𝑚 ≥ 12 𝑛(𝑘 − 1) + 12 𝑟 (𝑘 − 𝑟), where 𝑛 ≡ 𝑟 (mod 𝑘) and 0 ≤ 𝑟 < 𝑘. Proof Suppose, to the contrary, that the lemma is false, and let 𝐺 = (𝑉, 𝐸) be a counterexample of minimum size. The graph 𝐺 is edge-minimal with respect to the property that every vertex is in a 𝑘-clique. Every vertex of 𝐺 has degree at least 𝑘 − 1. Let 𝐴 be the set of vertices of 𝐺 of degree exactly 𝑘 − 1. Since every vertex of 𝐺 is in a 𝑘-clique, there is a collection 𝑉1 , 𝑉2 , . . . , 𝑉 𝑝 of 𝑘-element distinct subsets of 𝑉 each of which induces a clique in 𝐺 and whose union is 𝑉. Let 𝐴𝑖 = 𝐴 ∩ 𝑉𝑖 for 𝑖 ∈ [ 𝑝] and let 𝑎 𝑖 = | 𝐴𝑖 |. We note that 𝐴𝑖 is the set of those vertices in 𝑉𝑖 which are not contained in any
other set 𝑉 𝑗 . If 𝑝 = 1, then 𝐺 = 𝐾 𝑘 , and so 𝑛 = 𝑘 and 𝑟 = 0, implying that 𝑚 = 𝑛2 = 12 𝑛(𝑛 − 1) = 12 𝑛(𝑘 − 1) + 12 𝑟 (𝑘 − 𝑟), contradicting the supposition that 𝐺 is a counterexample. Hence, 𝑝 ≥ 2. We now sequentially consider the sets 𝑉𝑖 for 𝑖 ≥ 2. By the minimality of 𝐺, the set 𝐴𝑖 is nonempty. Suppose that 𝐴𝑖 ≠ 𝑉𝑖 . Let 𝐴1,𝑖 be an arbitrary set of 𝑘 − 𝑎 𝑖 vertices that belong to the set 𝑉1 . Further, let 𝐺 ′ be the graph obtained from 𝐺 by deleting the 𝑎 𝑖 (𝑘 − 𝑎 𝑖 ) edges between 𝐴𝑖 and 𝑉𝑖 \ 𝐴𝑖 and any edge in 𝐺 [𝑉𝑖 \ 𝐴𝑖 ] joining two vertices no longer in a common 𝑉 𝑗 , and adding the 𝑎 𝑖 (𝑘 − 𝑎 𝑖 ) edges between 𝐴𝑖 and 𝐴1,𝑖 . Note that in the resulting graph 𝐺 ′ , 𝑚(𝐺 ′ ) ≤ 𝑚(𝐺). Furthermore, every vertex of 𝐺 ′ is in a 𝑘-clique. Let 𝑉𝑖′ = 𝐴𝑖 ∪ 𝐴1,𝑖 for 𝑖 ≥ 2 and let 𝑉1′ = 𝑉1 . Thus, 𝑉1′ , 𝑉2′ , . . . , 𝑉 𝑝′ is a collection of 𝑘-element distinct subsets of 𝑉 each of which induces a clique in 𝐺 and whose union is 𝑉, and such that for every 𝑖 ≥ 2, the set 𝑉𝑖′ is either disjoint from the other sets 𝑉 𝑗′ or it overlaps with the 𝑘 − 𝑎 𝑖 vertices in 𝐴1,𝑖 of 𝑉1′ , but is otherwise disjoint in the sense that 𝑉𝑖′ ∩ 𝑉 𝑗′ ⊆ 𝑉1′ for all 𝑖 and 𝑗, where 2 ≤ 𝑖 < 𝑗 ≤ 𝑝. Hence, we may assume for the collection {𝑉1 , 𝑉2 , . . . , 𝑉 𝑝 } of 𝑘-element subsets in the original graph 𝐺, that the clique 𝐺 [𝑉𝑖 ], for 𝑖 ≥ 2, consists of 𝐴𝑖 and the first 𝑘 − 𝑎 𝑖 vertices of 𝑉1 . Thus, for every 𝑖 ≥ 2, the set 𝑉𝑖 is either disjoint from the other sets 𝑉 𝑗 , or it overlaps the first 𝑘 − 𝑎 𝑖 vertices of 𝑉1 , but is otherwise disjoint in the sense that 𝑉𝑖 ∩ 𝑉 𝑗 ⊆ 𝑉1 for all 𝑖 and 𝑗, where 2 ≤ 𝑖 < 𝑗 ≤ 𝑝. Suppose that two sets 𝑉𝑖 and 𝑉 𝑗 both intersect 𝑉1 for some 𝑖 and 𝑗, where 2 ≤ 𝑖 < 𝑗 ≤ 𝑝. Suppose that 𝑎 𝑖 + 𝑎 𝑗 ≤ 𝑘. In this case, we replace both sets 𝑉𝑖 and 𝑉 𝑗 with a single set 𝑉𝑖′ which consists of 𝐴𝑖 ∪ 𝐴 𝑗 and the first 𝑘 − 𝑎 𝑖 − 𝑎 𝑗 vertices of 𝑉1 . In other words, we let 𝐺 ′ be the graph obtained from 𝐺 by deleting the 𝑎 𝑖 (𝑘 − 𝑎 𝑖 )

Section 8.4. Independent Domination and Size

257

edges between 𝐴𝑖 and 𝑉𝑖 \ 𝐴𝑖 , deleting the 𝑎 𝑗 (𝑘 − 𝑎 𝑗 ) edges between 𝐴 𝑗 and 𝑉𝑖 \ 𝐴 𝑗 , adding the 𝑎 𝑖 𝑎 𝑗 edges between 𝐴𝑖 and 𝐴 𝑗 , and adding the (𝑎 𝑖 + 𝑎 𝑗 ) (𝑘 − 𝑎 𝑖 − 𝑎 𝑗 ) edges between 𝐴𝑖 ∪ 𝐴 𝑗 and the first 𝑘 − 𝑎 𝑖 − 𝑎 𝑗 vertices of 𝑉1 . In this case, we retain the property that every vertex of 𝐺 ′ is in a 𝑘-clique. However, the size of the resulting graph 𝐺 ′ is strictly less than the size of 𝐺, noting that 𝑚(𝐺 ′ ) = 𝑚 − 𝑎 𝑖 (𝑘 − 𝑎 𝑖 ) − 𝑎 𝑗 (𝑘 − 𝑎 𝑗 ) + 𝑎 𝑖 𝑎 𝑗 + (𝑎 𝑖 + 𝑎 𝑗 ) (𝑘 − 𝑎 𝑖 − 𝑎 𝑗 ) = 𝑚 − 𝑎𝑖 𝑎 𝑗 < 𝑚, contradicting the minimality of 𝐺. Hence, 𝑎 𝑖 + 𝑎 𝑗 > 𝑘. We now replace both sets 𝑉𝑖 and 𝑉 𝑗 with a set 𝑉𝑖′ that consists of 𝑘 vertices of 𝐴𝑖 ∪ 𝐴 𝑗 and a set 𝑉 𝑗′ that consists of the remaining 𝑎 𝑖 + 𝑎 𝑗 − 𝑘 vertices of 𝐴𝑖 ∪ 𝐴 𝑗 , together with the first 2𝑘 − 𝑎 𝑖 − 𝑎 𝑗 vertices of 𝑉1 . In other words, we let 𝐺 ′ be the graph obtained from 𝐺 by deleting the 𝑎 𝑖 (𝑘 − 𝑎 𝑖 ) edges between 𝐴𝑖 and 𝑉𝑖 \ 𝐴𝑖 , deleting the 𝑎 𝑗 (𝑘 − 𝑎 𝑗 ) edges between 𝐴 𝑗 and 𝑉 𝑗 \ 𝐴 𝑗 , adding the 𝑎 𝑖 (𝑘 − 𝑎 𝑖 ) edges between 𝐴𝑖 and a selected set 𝐴 𝑗,1 of 𝑘 − 𝑎 𝑖 vertices in 𝐴 𝑗 , adding the (𝑎 𝑖 + 𝑎 𝑗 − 𝑘) (2𝑘 − 𝑎 𝑖 − 𝑎 𝑗 ) edges between 𝑎 𝑖 + 𝑎 𝑗 − 𝑘 vertices in 𝐴 𝑗 \ 𝐴 𝑗,1 and the first 2𝑘 − 𝑎 𝑖 − 𝑎 𝑗 vertices of 𝑉1 , and deleting the (𝑎 𝑖 + 𝑎 𝑗 − 𝑘) (𝑘 − 𝑎 𝑖 ) edges between 𝐴 𝑗,1 and 𝐴 𝑗 \ 𝐴 𝑗,1 . In this case, we retain the property that every vertex of 𝐺 ′ is in a 𝑘-clique. However, the size of the resulting graph 𝐺 ′ is strictly less than the size of 𝐺, noting that 𝑚(𝐺 ′ ) = 𝑚 − 𝑎 𝑖 (𝑘 − 𝑎 𝑖 ) − 𝑎 𝑗 (𝑘 − 𝑎 𝑗 ) − (𝑎 𝑖 + 𝑎 𝑗 − 𝑘) (𝑘 − 𝑎 𝑖 ) + 𝑎 𝑖 (𝑘 − 𝑎 𝑖 ) + (𝑎 𝑖 + 𝑎 𝑗 − 𝑘) (2𝑘 − 𝑎 𝑖 − 𝑎 𝑗 ) = 𝑚 − (𝑘 − 𝑎 𝑖 ) (𝑘 − 𝑎 𝑗 ) < 𝑚, once again, contradicting the minimality of 𝐺. As a consequence of Lemma 8.26, we have the following result. Theorem 8.27 ([215]) If 𝐺 is a graph of order 𝑛 and size 𝑚 with 𝑖(𝐺) = 𝑘, where 𝑛 ≡ 𝑟 (mod 𝑘) and 0 ≤ 𝑟 < 𝑘, then
𝑚 ≤ 𝑛2 − 12 𝑛(𝑘 − 1) − 12 𝑟 (𝑘 − 𝑟). Proof Let 𝐺 be a graph of order 𝑛 and size 𝑚 with 𝑖(𝐺) = 𝑘, where 𝑛 ≡ 𝑟 (mod 𝑘) and 0 ≤ 𝑟 < 𝑘. If a vertex 𝑣 of 𝐺 belongs to a maximal independent set 𝐼 𝑣 of cardinality less than 𝑘, then 𝑖(𝐺) ≤ |𝐼 𝑣 | < 𝑘, a contradiction. Hence, every vertex of 𝐺 is in a (maximal) independent set of cardinality at least 𝑘. Therefore, in the complement 𝐺 of 𝐺, every vertex is in a 𝑘-clique. Thus, by Lemma 8.26, we have

𝑚 = 𝑛2 − 𝑚(𝐺) ≤ 𝑛2 − 12 𝑛(𝑘 − 1) − 12 𝑟 (𝑘 − 𝑟), as desired.

258

Chapter 8. Bounds in Terms of Size

8.5 Summary In this chapter, we presented relationships between the size of a graph and its domination, total domination, and independent domination numbers. This chapter concludes a sequence of four chapters on domination bounds, including general bounds for the three core domination numbers, bounds in terms of minimum degree and order, probabilistic bounds, and bounds in terms of size and order.

Chapter 9

Efficient Domination in Graphs 9.1

Introduction

In this chapter, we consider the concept of efficient domination, that is, the concept of dominating every vertex exactly once. An efficient dominating set 𝑆 ⊆ 𝑉 in a graph 𝐺 = (𝑉, 𝐸) is a dominating set with the additional property that the closed neighborhood N[𝑣] of every vertex 𝑣 ∈ 𝑉 contains exactly one vertex in 𝑆. It should be noted at the outset that not every graph has an efficient dominating set, the cycles 𝐶4 and 𝐶5 being two small examples. Thus, the study of efficient domination in graphs focuses primarily on families of graphs each member of which has an efficient dominating set, and then algorithms for finding such sets, or on families of graphs for which it can be determined in polynomial time which members do and do not have efficient dominating sets. The main focus of this chapter is on two types of efficient domination in graphs, namely, efficient domination and efficient total domination. Also, perfect domination will be discussed.

9.1.1

Efficient Dominating Sets

We can formally define an efficient dominating set in a graph 𝐺 in several different ways. Here is a standard way. Definition 9.1 A set 𝑆 ⊆ 𝑉 in a graph 𝐺 = (𝑉, 𝐸) is an efficient dominating set if for every vertex 𝑣 ∈ 𝑉, we have |N[𝑣] ∩ 𝑆| = 1, or equivalently, the closed neighborhood N[𝑣] contains exactly one vertex in 𝑆. For brevity, we will call an efficient dominating set 𝑆 an efficient set, and a graph 𝐺 having an efficient dominating set an efficient graph.  If 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } is an efficient set of 𝐺, then by definition N[𝑣 1 ], N[𝑣 2 ], . . . , N[𝑣 𝑘 ] is a partition of 𝑉 into 𝑘 pairwise disjoint closed neighborhoods, we will call this an efficient partition. An efficient partition need not be unique, for example, © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_9

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 N[𝑣 1 ], N[𝑣 4 ] and N[𝑣 2 ], N[𝑣 5 ] are two distinct efficient partitions of the path 𝑃5 : 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 . The observation that neither the cycle 𝐶4 nor the cycle 𝐶5 is an efficient graph immediately partitions all graphs into two classes: (i) efficient graphs, and (ii) those not having an efficient dominating set, called inefficient graphs. The fact that not all graphs are efficient immediately raises the question of the complexity of the following decision problem, to be discussed later in this chapter. 

EFFICIENT DOMSET (ED)

Instance: Graph 𝐺 = (𝑉, 𝐸) Question: Does 𝐺 have an efficient (dominating) set? Observation 9.2 If 𝑆 is an efficient set of a graph 𝐺, then for any two vertices 𝑢, 𝑣 ∈ 𝑆, 𝑑 (𝑢, 𝑣) > 2. Thus, an efficient set 𝑆 is both an independent set and a packing. From this observation, it follows that every efficient set is an independent dominating set, sometimes called an independent perfect dominating set. In 1988 Bange et al. [56] published what many graph theorists, as distinct from coding theorists, consider to be the first paper on efficient domination in graphs. But in this paper the authors state that the concept appeared in a Sandia Laboratories technical report ten years earlier by the same authors [55]. As we shall see later in this chapter, the concept of efficient domination was introduced from the aspect of coding theory even earlier in 1973. As originally noted in [56], every efficient set of a graph 𝐺 is a 𝛾-set of 𝐺. Proposition 9.3 ([56]) For any graph 𝐺, if 𝑆 and 𝑆 ′ are two distinct efficient sets, then |𝑆| = |𝑆 ′ | = 𝛾(𝐺). ′ Proof Let 𝑆 and 𝑆 ′ be two distinct efficient sets, and assume that  |𝑆| < |𝑆 |. Let 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } and consider the efficient partition 𝜋 = N[𝑣 1 ], N[𝑣 2 ], . . . , N[𝑣 𝑘 ] . By the Pigeonhole Principle, if |𝑆 ′ | > |𝑆|, there must be two vertices 𝑢 and 𝑣 in 𝑆 ′ that belong to the same closed neighborhood in 𝜋. But this means that 𝑑 (𝑢, 𝑣) ≤ 2, a contradiction. If |𝑆| is an efficient dominating set, then by definition 𝛾(𝐺) ≤ |𝑆|. But since every 𝛾-set of 𝐺 must contain at least one vertex in each closed neighborhood, then 𝛾(𝐺) ≥ |𝑆|. Thus, |𝑆| = 𝛾(𝐺).

9.1.2

Efficient Total Dominating Sets

Definition 9.4 A set 𝑆 ⊆ 𝑉 is an efficient total dominating set, also called an open efficient dominating set, of a graph 𝐺 = (𝑉, 𝐸) if for every vertex 𝑣 ∈ 𝑉, |N(𝑣) ∩ 𝑆| = 1, or equivalently, the open neighborhood N(𝑣) of 𝑣 contains exactly one vertex in 𝑆. For brevity, we will call an efficient total dominating set a total efficient set.  If 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } is a total efficient set, then by definition N(𝑣 1 ), N(𝑣 2 ), . . . , N(𝑣 𝑘 ) is a partition of 𝑉 into 𝑘 pairwise disjoint open neighborhoods, called a

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total efficient partition. Notice that if 𝑆 is a total efficient set, then the subgraph 𝐺 [𝑆] induced by 𝑆 is a disjoint union of 𝐾2 subgraphs. For a simple illustration, consider the  path 𝑃6 : 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑣 6 with total efficient set 𝑆 = {𝑣 1 , 𝑣 2 , 𝑣 5 , 𝑣 6 }, where N(𝑣 1 ),  N(𝑣 2 ), N(𝑣 5 ), N(𝑣 6 ) = {𝑣 2 }, {𝑣 1 , 𝑣 3 }, {𝑣 4 , 𝑣 6 }, {𝑣 5 } is a partition of 𝑉 (𝑃6 ) into four pairwise disjoint open neighborhoods. Notice in this case that the induced subgraph 𝐺 [𝑆] is a disjoint union of two 𝐾2 subgraphs. As with efficient sets, not every graph has a total efficient set and graphs having one are called total efficient graphs. Similar to efficient sets, every total efficient set of a graph 𝐺 is a 𝛾t -set of 𝐺.

9.1.3

Perfect Dominating Sets

Definition 9.5 A set 𝑆 ⊆ 𝑉 is a perfect dominating set of a graph 𝐺 = (𝑉, 𝐸) if for every vertex 𝑣 ∈ 𝑉 \ 𝑆, we have |N(𝑣) ∩ 𝑆| = 1, or equivalently, every vertex in 𝑉 \ 𝑆 has exactly one neighbor in 𝑆. As with efficient sets and total efficient sets, we will call a perfect dominating set a perfect set. Definition 9.6 A set 𝑆 ⊆ 𝑉 is a perfect total dominating set of a graph 𝐺 = (𝑉, 𝐸) if 𝑆 is a total dominating set and every vertex in 𝑉 \ 𝑆 has exactly one neighbor in 𝑆. A perfect total dominating set is called a total perfect set. Notice that by definition, every (total) efficient set is automatically a (total) perfect set, but not conversely. In other words, if 𝑆 is a perfect set, then 𝑆 is not necessarily an independent set or a packing.

9.1.4 Examples Consider the 4-cycle 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 1 . This cycle does not have an efficient set, but the set 𝑆 = {𝑣 1 , 𝑣 2 } is both a total efficient and a perfect set. By comparison, the 5-cycle 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑣 1 does not have an efficient set, nor does it have a total efficient set, but the set 𝑆 = {𝑣 1 , 𝑣 2 , 𝑣 3 } is both a perfect set and a total perfect set. The 6-cycle 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑣 6 𝑣 1 has an efficient set 𝑆 = {𝑣 1 , 𝑣 4 }, but does not have a total efficient set. The set 𝑆 = {𝑣 1 , 𝑣 4 } is also a perfect set, but so is the set 𝑆 ′ = {𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 }. This shows that a graph 𝐺 can have perfect sets of different cardinalities, while every efficient set in a graph 𝐺 must have the same cardinality, namely 𝛾(𝐺). The seven graphs in Figure 9.1 illustrate the distinctions between efficient, total efficient, and perfect sets. In an efficient set 𝑆, no two vertices in 𝑆 can be within distance 2 of each other, while in total efficient sets 𝑆, the vertices in 𝑆 appear in adjacent pairs, but no vertex in one pair is within distance 2 of a vertex in another pair. As previously mentioned for perfect domination, there are no constraints on the vertices within a perfect set; indeed, in the example given in Figure 9.1, they induce a connected subgraph. In closing this section, we refer the reader to the 2015 survey by Klostermeyer [529] of types of efficient domination. We should point out that in the literature different terminology has been used for different types of efficient dominating sets. For example, Fellows and Hoover [290] used the term “perfect dominating” for “efficient dominating,” “weakly perfect dominating” for “efficient total dominating,” and “semiperfect dominating” for “perfect dominating.”

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efficient

total efficient

total efficient

efficient

total efficient

perfect

total efficient

Figure 9.1 Efficient, total efficient, and perfect sets

9.2

Efficient Domination

In 1973 Biggs [76] wrote what is generally considered to be the first paper on efficient domination in graphs, although it was written from a coding theory perspective. His opening paragraph is the following: The problem of the existence of 𝑒-error correcting perfect codes of block length 𝑚 over GF(𝑞) is set in the vector space 𝑉 (𝑚, 𝑞), endowed with the Hamming metric; we shall refer to this as the classical perfect code question. It is possible to replace the vector space by a graph Γ(𝑚, 𝑞), whose vertices are vectors and whose edges join vectors which differ in precisely one coordinate. In fact, there is an analogous graph for all natural numbers 𝑞, not just the prime powers, so that we have a slight gain in generality. Using graph theory terminology, Biggs [76] defined a perfect 𝑒-code as follows. Let 𝐺 = (𝑉, 𝐸) be a connected graph. For each non-negative integer 𝑒 and each  vertex 𝑢 ∈ 𝑉, define the distance-𝑒 neighborhood of 𝑢 to be the set N𝑒 [𝑢] = 𝑣 : 𝑑 (𝑢, 𝑣) ≤ 𝑒 . A perfect 𝑒-code is a subset 𝑆 ⊆ 𝑉 such that the sets N𝑒 [𝑣], for all 𝑣 ∈ 𝑆, form a partition of 𝑉. Any connected graph 𝐺 therefore always has a perfect 0-code (let 𝑆 = 𝑉) and a perfect 𝑑-code where 𝑑 = diam(𝐺). These are called the two trivial codes. Hence, an efficient dominating set is a perfect 1-code. A classic example of perfect 1-codes occurs in the 𝑛-cubes 𝑄 𝑛 , the 2𝑛 vertices of which correspond 1-to-1 with the set of 2𝑛 𝑛-tuples of zeroes and ones, where two 𝑛-tuples are adjacent if and only if they differ in precisely one position, called Hamming distance-1. For 𝑛-cubes, a perfect 1-code, that is, an efficient dominating set 𝑆, corresponds to a set of code words which are to be transmitted from one location to another. If in transmission a single error occurs, then a bit string like 101 might be received as either 001 or 111 or 100. Consider the 3-cube 𝑄 3 with vertices: 𝑣 0 = (0, 0, 0), 𝑣 1 = (0, 0, 1), 𝑣 2 = (0, 1, 0), . . . , 𝑣 7 = (1, 1, 1). The set 𝑆 = {𝑣 0 , 𝑣 7 } is an efficient set of 𝑄 3 , since N[𝑣 0 ] = {𝑣 0 , 𝑣 1 ,

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𝑣 2 , 𝑣 4 } and N[𝑣 7 ] = {𝑣 3 , 𝑣 5 , 𝑣 6 , 𝑣 7 }. Thus, if in transmitting the code word (0, 0, 0) a single error occurs, and either (0, 0, 1) or (0, 1, 0) or (1, 0, 0) is received, it can be corrected uniquely to (0, 0, 0). It is well known in coding theory that the 𝑛-cube 𝑄 𝑛 has an efficient set if and only if 𝑛 = 2 𝑘 − 1, for some positive integer 𝑘. Thus, the 𝑛-cube is efficient if and only if 𝑛 = 2 𝑘 − 1. By Proposition 9.3, an efficient dominating set in 𝐺 has cardinality 𝛾(𝐺). In 2012 Brandstädt et al. [103] generalized this as follows, where N = {1, 2, . . .} is the set of positive integers. For a graph 𝐺 = (𝑉, 𝐸), define the following weighted Í vertex function w : 𝑉 → N by w(𝑣) = deg(𝑣) + 1. For 𝑆 ⊆ 𝑉, define w(𝑆) = 𝑣 ∈𝑆 w(𝑣). For any graph 𝐺 = (𝑉, 𝐸), the graph 𝐺 2 = (𝑉, 𝐸 2 ) is the graph in which 𝑢𝑣 ∈ 𝐸 2 if and only if 𝑑 (𝑢, 𝑣) ≤ 2 in 𝐺. Proposition 9.7 ([103]) For any graph 𝐺 = (𝑉, 𝐸) and corresponding weight function w : 𝑉 → N, and any subset 𝑆 ⊆ 𝑉, the following hold: (a) If 𝑆 is an efficient set in 𝐺, then w(𝑆) ≥ |𝑉 |. (b) If 𝑆 is an independent set in 𝐺 2 , then w(𝑆) ≤ |𝑉 |. Proposition 9.8 ([103]) For any graph 𝐺 = (𝑉, 𝐸) and corresponding weight function w : 𝑉 → N, the following are equivalent for any subset 𝑆 ⊆ 𝑉: (a) 𝑆 is an efficient set. (b) 𝑆 is a minimum weight dominating set in 𝐺 and w(𝑆) = |𝑉 |. (c) 𝑆 is a maximum weight independent set in 𝐺 2 and w(𝑆) = |𝑉 |.  Proof (a) ⇒ (b): If 𝑆 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } is an efficient set in 𝐺, then N[𝑣 1 ], N[𝑣 2 ], . . . , N[𝑣 𝑘 ] is a partition of 𝑉 and hence w(𝑆) = |𝑉 |. Also, by Proposition 9.7, there is no efficient set 𝑆 ′ ⊆ 𝑉 with w(𝑆 ′ ) < w(𝑆) = |𝑉 |. (b) ⇒ (c): If 𝑆 is a minimum weight dominating set in 𝐺 with w(𝑆) = |𝑉 | and the closed neighborhoods of the vertices in 𝑆 partition 𝑉, then 𝑆 is a maximum independent set in 𝐺 2 with w(𝑆) = |𝑉 |, since, by Proposition 9.7, there is no independent set 𝑆 ′ ⊂ 𝑉 in 𝐺 2 with w(𝑆 ′ ) > w(𝑆) = |𝑉 |. (c) ⇒ (b): If 𝑆 is a maximum independent set in 𝐺 2 with w(𝑆) = |𝑉 |, then the closed neighborhoods of the vertices in 𝑆 partition 𝑉, and thus, 𝑆 is an efficient set in 𝐺.

9.2.1

Efficient Graphs

It is easy to see that every graph with domination number 1 is efficient. For other simple examples, we note that all paths are efficient, while a cycle 𝐶𝑛 , for 𝑛 ≥ 3, is efficient if and only if 𝑛 ≡ 0 (mod 3). In 1986 Harary and Livingston [385] characterized efficient trees in terms of forbidden subgraphs. In 2000 Cockayne et al. [184] gave a different characterization of efficient trees in terms of vertex subsets that must be contained in all minimum dominating sets and in all minimum independent dominating sets in trees. In 1990 Livingston and Stout [568] studied the problem of finding efficient sets in a variety of families of graphs considered in the study of parallel interconnection

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networks. For trees, directed acyclic graphs, and series-parallel graphs, the authors presented a linear algorithm for deciding if an efficient set exists, and if it does, the algorithm constructs one. For 2- and 3-dimensional grid graphs, toroidal graphs, hypercubes 𝑄 𝑛 , and cube-connected paths, they characterized which graphs have efficient sets and they characterized their structure. The authors also studied efficient sets in higher-dimensional grids, cube-connected cycles, and de Bruijn graphs. In 1993 Clark [178] studied the probabilities that a random graph 𝐺 ∈ G(𝑛, 𝑝) is efficient. He showed that for a wide range of probabilities 𝑝 that an edge exists between any two vertices, almost every graph in G(𝑛, 𝑝) is inefficient, that is, has no efficient set. In addition, he showed that almost every tree in the set T𝑛 of all trees of order 𝑛 is inefficient.

9.2.2 Efficient Grid Graphs and Efficient Toroidal Graphs  An 𝑚 × 𝑛 grid graph 𝐺 𝑚,𝑛 has a vertex set 𝑉 = (𝑖, 𝑗) : 𝑖 ∈ [𝑚], 𝑗 ∈ [𝑛] , where (𝑖, 𝑗) is adjacent to (𝑘, 𝑙) if 𝑖 = 𝑘 and | 𝑗 − 𝑙 | = 1 or 𝑗 = 𝑙 and |𝑖 − 𝑘 | = 1. We often refer to a grid graph as simply a grid. For a fixed value of 𝑖, the set of vertices of the form (𝑖, 𝑗), 1 ≤ 𝑗 ≤ 𝑛, is called the 𝑖 th row of 𝐺 𝑚,𝑛 , and for a fixed value of 𝑗, the set of vertices of the form (𝑖, 𝑗), 1 ≤ 𝑖 ≤ 𝑚, is called the 𝑗 th column of 𝐺 𝑚,𝑛 . Thus, the vertex (𝑖, 𝑗) is placed in the 𝑖 th row and 𝑗 th column of the grid. For example, in Figure 9.2(a), an efficient set consists of the highlighted vertices (1, 1), (2, 3), and (1, 5); while in Figure 9.2(b), an efficient set consists of the highlighted vertices (1, 2), (2, 4), (3, 1), and (4, 3), where we adopt the convention that vertex (1, 1) appears in the lower left corner of the grid. While this way of labeling and drawing is commonly used by those who do research in grid graphs, it is clear that a grid graph is the Cartesian product 𝐺 𝑚,𝑛 = 𝑃𝑚 □ 𝑃𝑛 , where sometimes a different convention is used (as we shall see in Chapter 18). A toroidal graph, or just a torus, is a Cartesian product of the form 𝐶𝑚 □ 𝐶𝑛 . We use similar notation for the vertices of a torus 𝐶𝑚 □ 𝐶𝑛 as we do for a grid. That is, vertex (𝑖, 𝑗), where 𝑖 ∈ [𝑚] and 𝑗 ∈ [𝑛], is in the 𝑖 th row and 𝑗 th column of the torus. We note that the vertices (𝑖, 1), (𝑖, 2), . . . , (𝑖, 𝑛) induce a cycle 𝐶𝑛 for all 𝑖 ∈ [𝑚], while the vertices (1, 𝑗), (2, 𝑗), . . . , (𝑚, 𝑗) induce a cycle 𝐶𝑚 for all 𝑗 ∈ [𝑛].

(a) 𝑃2 □ 𝑃5 efficient

(b) 𝑃4 □ 𝑃4 efficient

Figure 9.2 Efficient sets in grids

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As previously mentioned, paths are efficient, that is, 𝐺 1,𝑛 = 𝑃𝑛 is efficient. Figure 9.2 illustrates the only two types of efficient sets in grids 𝐺 𝑚,𝑛 , for 𝑚, 𝑛 ≥ 2, which exist only in 2 × 𝑛 grids for odd 𝑛 and in the 4 × 4 grid, as shown by Livingston and Stout [568] in the following theorem. Theorem 9.9 ([568]) The grid 𝐺 𝑚,𝑛 = 𝑃𝑚 □ 𝑃𝑛 , for 𝑚, 𝑛 ≥ 2, has an efficient set if and only if either 𝑚 = 𝑛 = 4 or 𝑚 = 2 and 𝑛 = 2𝑘 + 1, where 𝑘 ≥ 1. Furthermore, the efficient set  𝑆 = (1, 2), (2, 4), (3, 1), (4, 3) for 𝐺 4,4 is unique up to isomorphism and the efficient set 
𝑆 ′ = 1 + 𝑖 (mod 2), 1 + 2𝑖 : 0 ≤ 𝑖 ≤ 𝑘 for 𝐺 2,2𝑘+1 is unique up to isomorphism. Proof It is easy to verify that the sets 𝑆 and 𝑆 ′ are efficient sets of 𝐺 4,4 and 𝐺 2,2𝑘+1 , respectively. Let 𝑆2 be an efficient dominating set of 𝐺 2,𝑛 . If (1, 1) is not in 𝑆2 , then exactly one of (1, 2) and (2, 1) must be in 𝑆2 . If (1, 2) is in 𝑆2 , then since 𝑆2 is an efficient set, none of (2, 1), (2, 2), and (1, 1) is in 𝑆2 . But then (2, 1) cannot be efficiently dominated, a contradiction. If (2, 1) is in 𝑆2 , then, by symmetry, the set 𝑆2 will be isomorphic to an efficient set not containing (2, 1) but containing (1, 1). Therefore, we can assume, without loss of generality, that (1, 1) ∈ 𝑆2 . But if (1, 1) ∈ 𝑆2 , then all other vertices in 𝑆2 are uniquely determined. In particular, (2, 3) ∈ 𝑆2 , and it follows that 𝑛 ≡ 1 (mod 2) is odd. We proceed further with the following claim. Claim 9.9.1 No graph 𝐺 3,𝑛 has an efficient set for 𝑛 ≥ 3. Proof Suppose, to the contrary, that 𝑆3 is an efficient set of 𝐺 3,𝑛 , where 𝑛 ≥ 3. Since it is easy to see by inspection that 𝐺 3,3 does not have an efficient set, we can assume that 𝑛 ≥ 4. There are only three ways to dominate vertex (1, 1). Case 1. (1, 1) ∈ 𝑆3 . In this case, vertices (1, 2), (1, 3), (2, 1), (2, 2), and (3, 1) cannot be in 𝑆3 , else some vertex will be dominated twice. The only way to dominate vertex (3, 1) is to include vertex (3, 2) in 𝑆3 . But this implies that vertices (3, 3) and (2, 3) are not in 𝑆3 . The only way to dominate (2, 3) is with (2, 4) ∈ 𝑆3 . But now vertex (1, 3) can no longer be efficiently dominated. Case 2. (2, 1) ∈ 𝑆3 . This implies that (1, 1), (3, 1), (1, 2), (2, 2), (3, 2), and (2, 3) cannot be in 𝑆3 . Hence, (1, 3) and (3, 3) are in 𝑆3 in order to dominate (1, 2) and (3, 2), respectively, contradicting the efficiency of the set 𝑆3 since (2, 3) is dominated twice. Case 3. (1, 2) ∈ 𝑆3 . This implies that (1, 1), (2, 1), (2, 2), (3, 2), (2, 3), (1, 3), and (1, 4) cannot be in 𝑆3 , else some vertex will be dominated twice. This implies that (3, 1) ∈ 𝑆3 in order to dominate (2, 1). Hence, (3, 4) and (2, 4) are in 𝑆3 in order to dominate (3, 3) and (2, 3), respectively, contradicting the efficiency of the set 𝑆3 since (2, 4) and (3, 4) are adjacent. Since all three cases produce a contradiction and since exactly one of these three cases must occur, this completes the proof of Claim 9.9.1.

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By Claim 9.9.1, no graph 𝐺 3,𝑛 has an efficient set for 𝑛 ≥ 3. Let 𝑆 𝑚 be an efficient set in 𝐺 𝑚,𝑛 for 𝑚, 𝑛 ≥ 4. To dominate vertex (1, 1), exactly one of (1, 1), (1, 2), and (2, 1) is in 𝑆 𝑚 . By the symmetry of vertices (1, 2) and (2, 1), there are only two cases to consider, namely, (1, 1) ∈ 𝑆 𝑚 or (1, 2) ∈ 𝑆 𝑚 . If (1, 1) ∈ 𝑆 𝑚 , then (1, 2), (1, 3), (2, 1), (2, 2), and (3, 1) cannot be in 𝑆 𝑚 , else some vertex will be dominated twice. Thus, only vertices (3, 2) or (2, 3) are available to dominate vertex (2, 2). By symmetry, we only need to consider one of these two cases, so assume that (2, 3) ∈ 𝑆 𝑚 . This implies that vertex (3, 2) ∉ 𝑆 𝑚 and (3, 3) ∉ 𝑆 𝑚 . But this means that both (4, 1) and (4, 2) are in 𝑆 𝑚 to dominate (3, 1) and (3, 2), respectively, contradicting the efficiency of the set 𝑆 𝑚 since (4, 1) and (4, 2) are adjacent. If (1, 2) ∈ 𝑆 𝑚 , then (1, 1), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), and (3, 2) cannot be in 𝑆3 , else some vertex will be dominated twice. This forces vertex (3, 1) to be in 𝑆 𝑚 in order to dominate vertex (2, 1). This implies that (3, 3), (4, 1), and (4, 2) are not in 𝑆 𝑚 . This in turn implies that (2, 4) ∈ 𝑆 𝑚 in order to dominate (2, 3). Vertex (4, 3) must then be in 𝑆 𝑚 in order to efficiently dominate (3, 3). Now if 𝑚 = 𝑛 = 4, then 𝑆 𝑚 is an efficient set in 𝐺 4,4 , as illustrated in Figure 9.2(b). If 𝑛 = 5, then vertex (𝑖, 5), for 𝑖 ∈ [4], is not in 𝑆 𝑚 , else some vertex will be dominated twice. Hence, vertices (1, 5) and (3, 5) cannot be efficiently dominated by 𝑆 𝑚 . If 𝑛 ≥ 6, then vertex (1, 6) must be in 𝑆 𝑚 in order to dominate (1, 5). And this then means that vertex (3, 5), which cannot be in 𝑆 𝑚 , can only be dominated by (3, 6), again contradicting the efficiency of the set 𝑆 𝑚 . It is easy to see that the two vertices (1, 1, 1) and (2, 2, 2) form an efficient set in the 3-dimensional grid graph, known as the cube 𝑄 3 . It is somewhat surprising that there are no other efficient 3-dimensional grid graphs. Theorem 9.10 ([568]) The 3-dimensional grid graph 𝐺 𝑚1 ,𝑚2 ,𝑚3 has an efficient set if and only if 𝑚 1 = 𝑚 2 = 𝑚 3 = 2. We conclude this subsection with a characterization of efficient toroidal graphs given by Klavžar and Seifter [526] in 1995. Theorem 9.11 ([526]) For 𝑚 ≥ 3 and 𝑛 ≥ 3, the torus 𝐶𝑚 □ 𝐶𝑛 has an efficient set if and only if 𝑚, 𝑛 ≡ 0 (mod 5). Figure 9.3 illustrates an efficient set in the torus 𝐶5 □ 𝐶5 .

9.2.3

Efficient Cube-connected Cycles

In 1993 Van Wieren et al. [729] studied efficient sets in cube-connected cycles, a family of cubic graphs having relatively small diameters and a regular structure making them useful models for parallel computer architectures. A cube-connected cycle 𝐶𝐶𝐶𝑛 is a graph formed from an 𝑛-cube 𝑄 𝑛 by replacing each of the 2𝑛 vertices of 𝑄 𝑛 with a cycle of order 𝑛, as illustrated in Figure 9.4. The existence of efficient sets in such graphs facilitates the design of efficient algorithms. Van Wieren et al. [729] gave a simple method of constructing efficient

Section 9.2. Efficient Domination

267

Figure 9.3 Efficient set (red vertices) in the torus 𝐶5 □ 𝐶5 (2, 011)

(1, 011)

(2, 111)

(3, 011)

(2, 010)

(1, 010)

(3, 111)

(2, 110)

(3, 010)

(1, 110)

(3, 110)

(2, 001)

(1, 001)

(2, 101)

(3, 001)

(2, 000)

(1, 000)

(3, 000)

(1, 111)

(1, 101)

(3, 101)

(2, 100)

(1, 100)

(3, 100)

Figure 9.4 Cube-connected cycle of dimension 3

sets in such graphs if they exist and proved that they do not exist otherwise. Efficient sets were shown to exist in cube-connected cycles of order 𝑘 for 𝑘 ≠ 5. For example, the set  (1, 000), (1, 111), (2, 011), (2, 100), (3, 010), (3, 101) is an efficient set of the cube-connected cycle 𝐶𝐶𝐶3 shown in Figure 9.4.

9.2.4

Efficient Vertex-transitive Graphs

A graph 𝐺 is vertex-transitive if for any two vertices 𝑢 and 𝑣 of 𝐺, there is an automorphism 𝑓 : 𝐺 → 𝐺 such that 𝑓 (𝑢) = 𝑣. In other words, a graph 𝐺 is

Chapter 9. Efficient Domination in Graphs

268

vertex-transitive if its automorphism group acts transitively on 𝑉 (𝐺). For a simple example, cycles are vertex-transitive. In 2012 Knor and Potočnik [532] studied efficient domination in cubic vertextransitive graphs. The authors considered the general problem: characterize vertextransitive graphs that have efficient sets. Noting that 2-regular graphs are efficient if and only if they are isomorphic to a disjoint union of cycles of lengths divisible by 3, they considered 3-regular vertex-transitive graphs. They presented the number # of cubic vertex-transitive graphs of a given order |𝑉 |, followed by the number #ED of these having an efficient set as shown in Table 9.1. |𝑉 |

4

1 # #ED 1

8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 2 1

4 2

4 3

7 11 6 10 12 12 7 32 10 16 38 26 12 37 11 2 6 2 9 3 6 1 23 2 8 4 25 2 19 1

Table 9.1 Number of efficient cubic vertex-transitive graphs The Möbius ladder 𝑀𝑛 is the cubic graph obtained from a cycle 𝐶2𝑛 by adding 𝑛 chords of a perfect matching which connect pairs of opposite vertices. One can note that the smallest Möbius ladders 𝑀2 and 𝑀3 are isomorphic to 𝐾4 and 𝐾3,3 , respectively. Theorem 9.12 ([532]) For 𝑚 ≥ 2, if 𝐺 is a connected cubic vertex-transitive graph of order 𝑛 = 2𝑚 , then 𝐺 does not have an efficient set if and only if 𝑚 ≥ 3 and 𝐺 = 𝑀2𝑚−1 . The authors also noted the following. Proposition 9.13 ([532]) For 𝑛 ≥ 3, a prism 𝐶𝑛 □ 𝐾2 is efficient if and only if 𝑛 ≡ 0 (mod 4). Proposition 9.14 ([532]) The Möbius ladder 𝑀𝑛 is efficient if and only if 𝑛 ≡ 2 (mod 4), or equivalently, if and only if |𝑉 (𝑀𝑛 )| ≡ 4 (mod 8).

9.2.5

Efficient Cayley Graphs

In 2001 Lee [556] studied efficient Cayley graphs. These graphs are defined as follows. Given a finite group Γ with an identity element 𝑒 (or 0) and an inverse-closed, symmetric subset 𝑆 of Γ not containing the identity element, that is, if 𝑎 ∈ 𝑆, then 𝑎 −1 ∈ 𝑆, the Cayley graph Cay(Γ, 𝑆) on Γ relative to the connection set 𝑆 is the graph with vertex set Γ such that vertices 𝑢 and 𝑣 are adjacent if and only if 𝑣𝑢 −1 ∈ 𝑆. This graph is regular of degree |𝑆|, and is connected if and only if 𝑆 is a generating set of Γ. In the special case when Γ = Z𝑛 , the additive group of integers modulo 𝑛, a Cayley graph Cay(Z𝑛 , 𝑆) on Z𝑛 is called a circulant. A graph 𝐺˜ is called a covering of 𝐺 with projection 𝑓 : 𝐺˜ → 𝐺 if there is a ˜ → 𝑉 (𝐺) such that, for any vertex 𝑣 ∈ 𝑉 (𝐺), 𝑓 | N( 𝑣) surjection 𝑓 : 𝑉 ( 𝐺) ˜ : N( 𝑣˜ ) →

Section 9.2. Efficient Domination

269

N(𝑣) is a bijection, where 𝑣˜ ∈ 𝑓 −1 𝑣. In addition, the projection 𝑓 : 𝐺˜ → 𝐺 is said to be an 𝑛-fold covering if 𝑓 is 𝑛-to-one. Theorem 9.15 ([556]) A Cayley graph of an abelian group is efficient if and only if it is a covering graph of a complete graph. As an application, Lee [556] proved the following. Theorem 9.16 ([556]) The following are equivalent: (a) The hypercube 𝑄 𝑛 is efficient. (b) 𝑄 𝑛 is a regular covering of the complete graph 𝐾𝑛+1 . (c) 𝑛 = 2𝑑 − 1 for some natural number 𝑑. In proving Theorems 9.15 and 9.16, Lee proved the following. Lemma 9.17 ([556]) If 𝑆1 , 𝑆2 , . . . , 𝑆 𝑛 are 𝑛 pairwise disjoint, efficient sets of a graph 𝐺, then the subgraph 𝐻 induced by 𝑆1 ∪ 𝑆2 ∪ · · · ∪ 𝑆 𝑛 is an 𝑚-fold covering graph of the complete graph 𝐾𝑛 , where 𝑚 = |𝑆𝑖 | for each 𝑖 ∈ [𝑛]. Proof It is easy to show that for any two disjoint efficient sets 𝑆𝑖 and 𝑆 𝑗 , the induced subgraph 𝐺 [𝑆𝑖 ∪ 𝑆 𝑗 ] is isomorphic to the Cartesian product 𝐾2 □ 𝐾 𝑚 , where 𝑚 = |𝑆𝑖 | = |𝑆 𝑗 |. This implies that for each vertex 𝑢 ∈ 𝑆𝑖 and each 𝑗 ≠ 𝑖, there exists a unique vertex 𝑣 ∈ 𝑆 𝑗 such that 𝑢 and 𝑣 are adjacent in the subgraph 𝐻 of 𝐺 induced by 𝑆1 ∪ 𝑆2 ∪ · · · ∪ 𝑆 𝑛 . Let 𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 be the vertices of a complete graph 𝐾𝑛 . Then the function 𝑓 : 𝐻 → 𝐾𝑛 defined by 𝑓 (𝑠) = 𝑣 𝑖 for each 𝑠 ∈ 𝑆𝑖 is a covering projection. Lemma 9.18 ([556]) If 𝑓 : 𝐺˜ → 𝐺 is a covering and 𝑆 is a perfect set of 𝐺, ˜ Moreover, if 𝑆 is independent, then 𝑓 −1 (𝑆) is then 𝑓 −1 (𝑆) is a perfect set of 𝐺. independent. Proof It follows from the definition of a covering projection 𝑓 that 𝑓 −1 (𝑆) is ˜ \ 𝑓 −1 (𝑆). Then 𝑓 ( 𝑣˜ ) = 𝑣 ∈ 𝑉 (𝐺) \ 𝑆. independent if 𝑆 is independent. Let 𝑣˜ ∈ 𝑉 ( 𝐺) Since 𝑆 is a perfect set, there exists a unique vertex 𝑠 ∈ 𝑆 such that 𝑣 and 𝑠 are adjacent. Since 𝑓 is a covering projection, there exists a vertex 𝑠˜ ∈ 𝑓 −1 (𝑠) such that 𝑣˜ and 𝑠˜ are adjacent. One can show that such a vertex 𝑠˜ is unique. Let 𝑠˜′ be a vertex in 𝑓 −1 (𝑆) such that 𝑣˜ and 𝑠˜′ are adjacent. Then 𝑓 ( 𝑣˜ ) = 𝑣 and 𝑓 ( 𝑠˜′ ) are adjacent in 𝐺. Since 𝑆 is a perfect set, 𝑓 ( 𝑠˜′ ) = 𝑠 = 𝑓 ( 𝑠˜). But since 𝑓 is a covering projection, 𝑠˜ = 𝑠˜′ . Theorem 9.19 ([556]) A graph 𝐺 is a covering of the complete graph 𝐾𝑛 if and only if 𝐺 has a vertex partition {𝑆1 , 𝑆2 , . . . , 𝑆 𝑛 } such that 𝑆𝑖 is an efficient set for all 𝑖 ∈ [𝑛]. Further results on efficient domination in Cayley graphs can be found in the 2019 survey by Tamizh Chelvam and Sivagami [701].

Chapter 9. Efficient Domination in Graphs

270

9.2.6 Efficient Circulant Graphs    A circulant graph 𝐶𝑛 ⟨𝐿⟩ with a given list 𝐿 ⊆ 1, 2, . . . , 21 𝑛 is a graph on 𝑛 ≥ 3 vertices in which the 𝑖 th vertex is adjacent to the (𝑖 + 𝑗) th and (𝑖 − 𝑗) th vertices for each 𝑗 in the list 𝐿 and where is taken modulo 𝑛. More precisely, if 𝐿 = {𝑠1 , 𝑠2 ,  addition  . . . , 𝑠𝑟 } ⊆ 1, 2, . . . , 12 𝑛 , then thecirculant graph 𝐶𝑛 ⟨𝐿⟩ is the graph with vertex set {𝑣 0 , 𝑣 1 , . . . , 𝑣 𝑛−1 } and edge set 𝑣 𝑖 𝑣 𝑖+ 𝑗 (mod 𝑛) : 𝑖 ∈ [𝑛 − 1] 0 and 𝑗 ∈ {𝑠1 , 𝑠2 , . . . , 𝑠𝑟 } . Equivalently, if 𝐿 is a subset of the finite cyclic group Z𝑛 not containing the identity element 0 and 𝐿 = −𝐿, then a circulant graph is a Cayley graph Cay(Z𝑛 , 𝐿) on Z𝑛 with respect to 𝐿. As such, circulant graphs are the only family of vertextransitive graphs for which the number of vertices can be prime (see Turner [719]). For example, the circulant graphs 𝐶8 ⟨1, 2⟩ and 𝐶10 ⟨1, 2, 3⟩ are shown in Figure 9.5(a) and (b), respectively.

(a) 𝐶8 ⟨1, 2⟩

(b) 𝐶10 ⟨1, 2, 3⟩

Figure 9.5 The circulant graphs 𝐶8 ⟨1, 2⟩ and 𝐶10 ⟨1, 2, 3⟩ Circulant graphs form an important class of topological structures in interconnection networks, having been widely used in telecommunication networks, VLSI design, and distributed computation, in part because of their symmetry, fault-tolerance, and efficient routing capabilities. In 2007 Obradović et al. [621] studied efficient sets in circulant graphs having only two chord lengths, called 2-chord circulants. Lemma 9.20 ([621]) If 𝐶𝑛 ⟨𝑠1 , 𝑠2 ⟩ is an efficient connected 4-regular graph such that gcd(𝑠1 , 𝑛) = 1 and/or gcd(𝑠2 , 𝑛) = 1, then |𝑠1 ± 𝑠2 | . 0 (mod 5) and 𝑠1 , 𝑠2 . 0 (mod 5) Lemma 9.21 ([621]) If 𝐶5𝑘 ⟨𝑠1 , 𝑠2 ⟩ is an efficient connected 4-regular graph such that gcd(𝑠1 , 𝑛) ≠ 1 and gcd(𝑠2 , 𝑛) ≠ 1, then |𝑠1 ± 𝑠2 | . 0 (mod 5) and 𝑠1 , 𝑠2 . 0 (mod 5). Theorem 9.22 ([621]) If 𝐺 = 𝐶𝑛 ⟨𝑠1 , 𝑠2 ⟩ is a connected 4-regular graph, then 𝐺 is efficient if and only if each of the following holds: (a) 𝑛 = 5𝑘 (b) |𝑠1 ± 𝑠2 | . 0 (mod 5) (c) 𝑠1 , 𝑠2 . 0 (mod 5). Proof If 𝐺 = 𝐶𝑛 ⟨𝑠1 , 𝑠2 ⟩ is a connected 4-regular graph and 𝐺 is efficient, then conditions (a), (b), and (c) follow from Lemmas 9.20 and 9.21 and the definition of efficient domination, which implies that 𝑛 ≡ 0 (mod 5).

Section 9.2. Efficient Domination

271

 Conversely, let 𝐷 = 0, 5, 10, . . . , 5(𝑖 − 1) and let 𝑢 ∈ 𝑉 \ 𝐷. Since 𝑠1 , 𝑠2 . 0 (mod 5), the set 𝐷 is an independent set in 𝐺. Vertex 𝑢 can be written as 𝑢 = 5𝑞 + 𝑟, for 0 ≤ 𝑞 ≤ (𝑖 − 1) and 1 ≤ 𝑟 ≤ 4. Since |𝑠1 ± 𝑠2 | . 0 (mod 5), the chord lengths 𝑠1 , 𝑠2 , −𝑠1 , −𝑠2 must be in four different congruence classes modulo 5 and none of these classes are 0 (mod 5) since 𝑠1 , 𝑠2 . 0 (mod 5). Therefore, exactly one chord length is in the same congruence class modulo 5 as −𝑟, and 𝑢 has a single neighbor in 𝐷. Hence, 𝐷 is an efficient set of 𝐺. The four graphs in Figure 9.6 show the four possible efficient sets in the 2-chord circulants of order 𝑛 = 10.

(a) 𝐶10 ⟨1, 2⟩

(b) 𝐶10 ⟨1, 3⟩

(c) 𝐶10 ⟨2, 4⟩

(d) 𝐶10 ⟨3, 4⟩

Figure 9.6 Efficient sets in 2-chord circulants

In a remark at the end of their paper, Obradović et al. [621] gave an example of an 8-regular circulant graph 𝐺 = 𝐶27 ⟨2, 7, 9, 11⟩ for which the sets in the form {𝑎, 𝑏, 𝑐}, where 𝑎 ∈ {0, 9, 18}, 𝑏 ∈ {3, 12, 21}, and 𝑐 ∈ {6, 15, 24}, are efficient sets of 𝐺. Note that some of these efficient sets contain vertices that are not equally spaced in 𝐺. In 2013 Reji Kumar and MacGillivray [656] considered the construction of infinitely many efficient circulant graphs in which the elements of an efficient set need not be equally spaced. They showed that if a circulant graph of sufficiently large degree has an efficient set 𝐷, then either the elements of 𝐷 are equally spaced in 𝐺 or 𝐺 = 𝐻 ◦ 𝐾𝑚 , where 𝐻 is a smaller efficient circulant graph. They also gave the following result. Theorem 9.23 ([656]) If 𝐹 = 𝐶𝑚𝑛 ⟨𝑇⟩, where 𝑇 ∪ {0} is a union of cosets of 𝑛Z𝑛 , then 𝐹 is efficient if and only if 𝐶𝑛 ⟨𝑇⟩ is efficient. The lexicographic product 𝐺 ◦ 𝐻 of two graphs 𝐺 and 𝐻 is the graph with 𝑉 (𝐺 ◦ 𝐻) = 𝑉 (𝐺) × 𝑉 (𝐻), where two vertices (𝑢, 𝑣) and (𝑤, 𝑥) are adjacent whenever (i) (𝑢, 𝑤) ∈ 𝐸 (𝐺) or (ii) 𝑢 = 𝑤 and (𝑣, 𝑥) ∈ 𝐸 (𝐻). In 2009 Taylor [705] proved the following result, which was also proven independently by Reji Kumar and MacGillivray [656]. Theorem 9.24 ([705]) If 𝐺 is an isolate-free graph and 𝐻 is any graph, then the lexicographic product 𝐺 ◦ 𝐻 is efficient if and only if 𝐺 is efficient and 𝐻 is efficient with 𝛾(𝐻) = 1.

272

Chapter 9. Efficient Domination in Graphs

9.2.7 Efficient Graphs with Efficient Complements In 1998 Harary et al. [384] characterized the coronas and the caterpillars having efficient sets. In addition, they characterized the family of graphs 𝐺 for which both 𝐺 and 𝐺 have efficient sets, as follows. Let F be the family of graphs containing the path 𝑃4 and every graph that can be obtained from the union of a path 𝑃4 : 𝑢 𝑥 𝑦 𝑣 and an arbitrary graph 𝐻 by adding edges such that each vertex of 𝐻 is adjacent to exactly one of 𝑢 and 𝑣 and to exactly one of 𝑥 and 𝑦. Equivalently, 𝐺 ∈ F if and only if 𝐺 has an induced path 𝑃4 : 𝑢 𝑥 𝑦 𝑣 such that each vertex in 𝑉 \ {𝑢, 𝑥, 𝑦, 𝑣} has exactly one neighbor in {𝑢, 𝑣} and exactly one neighbor in {𝑥, 𝑦}. We note that if 𝐺 ∈ F , then 𝐺 ∈ F . Theorem 9.25 ([384]) A graph 𝐺 and its complement 𝐺 are isolate-free efficient graphs if and only if 𝐺 ∈ F . Proof It is straightforward to see that if 𝐺 ∈ F , then 𝐺 is efficient. If 𝐺 ∈ F , then 𝐺 ∈ F , so both 𝐺 and 𝐺 are efficient. Let 𝐺 be a graph such that both 𝐺 and 𝐺 are isolate-free efficient graphs. Since 𝐺 has no isolated vertices, it follows that 𝛾(𝐺) ≥ 2. Similarly, 𝛾(𝐺) ≥ 2. Moreover, since the distance between any two vertices in an efficient set of 𝐺 (respectively 𝐺) is at least 3, diam(𝐺) ≥ 3 and diam(𝐺) ≥ 3. Note that any two vertices at distance 3 or more apart in 𝐺 form a dominating set of 𝐺, implying that 𝛾(𝐺) ≤ 2 and so 𝛾(𝐺) = 2. Similarly, 𝛾(𝐺) = 2.  Let {𝑢, 𝑣} be an efficient set of 𝐺. Then N𝐺 [𝑢], N𝐺 [𝑣] is a partition of the  vertices of 𝐺, implying that N𝐺 (𝑢), N𝐺 (𝑣) is a vertex partition in 𝐺. In other words, 𝑢𝑣 is a dominating edge of 𝐺, that is, an adjacent pair of vertices which forms a dominating set in a graph 𝐺. Similarly, for any efficient set {𝑥, 𝑦} of 𝐺, 𝑥𝑦 is a dominating edge of 𝐺. Since no vertex in N𝐺 [𝑢] dominates 𝑣 and no vertex in N𝐺 [𝑣] dominates 𝑢 in 𝐺, it follows that neither 𝑢 nor 𝑣 can be incident to a dominating edge in 𝐺. Hence, {𝑢, 𝑣} ∩ {𝑥, 𝑦} = ∅. Moreover, 𝑢 is adjacent to exactly one of 𝑥 and 𝑦, say 𝑥, in 𝐺. This implies that 𝑣 is not adjacent to 𝑥 but is adjacent to 𝑦 in 𝐺. Thus, 𝑢 𝑥 𝑦 𝑣 is an induced 𝑃4 in 𝐺.  Since {𝑥, 𝑦} is an efficient set of 𝐺, N𝐺 [𝑥], N𝐺 [𝑦] is a vertex partition of 𝐺 and so, N𝐺 (𝑥), N𝐺 (𝑦) is a vertex partition in 𝐺. Thus, every vertex in 𝑉 \ {𝑢, 𝑥, 𝑦, 𝑣} has exactly one neighbor in {𝑢, 𝑣} and exactly one neighbor in {𝑥, 𝑦} in 𝐺. Hence, 𝐺 ∈ F . The path 𝑃4 ∈ F is an example of an efficient self-complementary graph. For another example of a graph 𝐺 in F , see Figure 9.7, where the blue vertices form an efficient set in the graph 𝐺, while the red vertices form an efficient set in its complement 𝐺.

9.3

Efficient Total Domination

In 2002 Gavlas and Schultz [328] and in 2003 Gavlas et al. [329] studied efficient total domination in graphs and presented the following results.

Section 9.3. Efficient Total Domination

273

Figure 9.7 A graph 𝐺 such that both 𝐺 and 𝐺 are efficient

Theorem 9.26 ([328]) A path 𝑃𝑛 has a total efficient set if and only if 𝑛 . 1 (mod 4). Theorem 9.27 ([328]) A cycle 𝐶𝑛 has a total efficient set if and only if 𝑛 ≡ 0 (mod 4).

9.3.1

Total Efficient Trees

Gavlas and Schultz [328] characterized the class of trees having a total efficient set as follows. Let T denote the class of trees with total efficient sets 𝑆 that can be constructed from the tree 𝑇1 = 𝐾2 and set 𝑆1 = 𝑉 (𝐾2 ) by a sequence 𝑇1 , 𝑇2 , . . . , 𝑇𝑘 = 𝑇 of trees and total efficient sets 𝑆1 , 𝑆2 , . . . , 𝑆 𝑘 = 𝑆, where 𝑘 ≥ 1, and if 𝑘 ≥ 2, then the tree 𝑇𝑖+1 can be obtained from the tree 𝑇𝑖 by applying one of the following two operations for all 𝑖 ∈ [𝑘 − 1]: Operation 1. Attach a leaf to a vertex of 𝑆𝑖 , and let 𝑆𝑖+1 = 𝑆𝑖 . Operation 2. Add a path 𝑢 𝑣 𝑤 𝑥, where 𝑢 ∈ 𝑉 (𝑇𝑖 ) \ 𝑆𝑖 and 𝑣, 𝑤, and 𝑥 are new vertices not in 𝑇𝑖 , and let 𝑆𝑖+1 = 𝑆𝑖 ∪ {𝑤, 𝑥}. Note that 𝑢 𝑣 𝑤 𝑥 induces a path 𝑃4 in 𝑇𝑖+1 and that vertex 𝑢 must be dominated by a vertex in 𝑆𝑖 . Theorem 9.28 ([328]) A nontrivial tree 𝑇 has a total efficient set 𝑆 if and only if 𝑇 ∈ T. Proof We proceed by induction on the order 𝑛 ≥ 2 of a tree 𝑇. It is easy to see that all trees of order 𝑛 with 2 ≤ 𝑛 ≤ 6 having a total efficient set are in T . Assume that all nontrivial total efficient trees of order at most 𝑛′ are in T and let 𝑇 be a tree of order 𝑛 = 𝑛′ + 1 having a total efficient set 𝑆. We show that 𝑇 ∈ T . If the tree 𝑇 has a leaf 𝑣 not in 𝑆, then the tree 𝑇 ′ = 𝑇 − 𝑣 has a total efficient set 𝑆. Hence, by the inductive hypothesis, 𝑇 ′ ∈ T , and by applying Operation 1 to 𝑇 ′ , attaching the leaf 𝑣, we can construct 𝑇. Therefore, 𝑇 ∈ T . Hence, we may assume that every leaf in 𝑇 belongs to 𝑆. Thus, no vertex can have two leaf neighbors, implying that 𝑇 is not a star. Hence, 𝑇 has diameter at least 3. Let 𝑃 be a longest path in 𝑇 and let 𝑥, 𝑤, 𝑣, 𝑢 be the first four vertices of 𝑃. By our earlier assumptions, all leaves of 𝑇 belong to 𝑆. In particular, the vertex 𝑥 belongs to 𝑆. Therefore, vertex 𝑤 must also be in 𝑆. Hence, the vertices 𝑣 and 𝑢 cannot be in 𝑆. If deg(𝑤) > 2, then the vertex 𝑤 has a neighbor 𝑦 not on 𝑃. Since 𝑃 is a longest path, the vertex 𝑦 is a leaf and therefore belongs to the set 𝑆, which means that

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the vertex 𝑤 is dominated twice, once by 𝑥 and once by 𝑦, a contradiction. Hence, deg(𝑤) = 2. Suppose that deg(𝑣) > 2 and so the vertex 𝑣 has a neighbor 𝑧 not on 𝑃. If the vertex 𝑧 belongs to 𝑆, then the vertex 𝑣 is dominated twice, a contradiction. Hence, 𝑧 cannot be in 𝑆. Thus, the vertex 𝑧 cannot be a leaf, and therefore must have a neighbor 𝑧 ′ different from 𝑣. Since 𝑃 is a longest path, the vertex 𝑧 ′ is a leaf and therefore belongs to the set 𝑆. However, since 𝑧 ∉ 𝑆, the vertex 𝑧 ′ has no neighbor in 𝑆, a contradiction. Hence, deg(𝑣) = 2. By our earlier observations, {𝑤, 𝑥} ⊆ 𝑆 and neither 𝑣 nor 𝑢 belong to 𝑆. Further, 𝑥 is a leaf and deg(𝑤) = deg(𝑣) = 2. Deleting the vertices 𝑣, 𝑤, 𝑥 from 𝑇 results in a tree 𝑇 ′ having a total efficient set 𝑆 ′ = 𝑆 \ {𝑤, 𝑥}. By induction 𝑇 ′ ∈ T and therefore 𝑇 ∈ T by virtue of applying Operation 2 to 𝑇 ′ . Conversely, it follows the construction that 𝑇 = 𝑇𝑘 has a total efficient set.

9.3.2

Total Efficient Grid Graphs

In 2006 Klostermeyer and Goldwasser [530] characterized the class of all grid graphs 𝑃𝑚 □ 𝑃𝑛 having a total perfect code, that is, a total efficient set. This extended the work of Gavlas and Schultz [328] for 𝑚 = 1 and the work of Cowen et al. [204], who solved this problem for odd-by-odd grids. Theorem 9.29 ([204]) For 𝑚, 𝑛 > 1, if 𝑚 and 𝑛 are both odd, then 𝑃𝑚 □ 𝑃𝑛 is not total efficient. Theorem 9.30 ([530]) The following hold: (a) The 1 × 𝑛 grid is total efficient if and only if 𝑛 . 1 (mod 4). (b) For 𝑚, 𝑛 > 1, the grid 𝑃𝑚 □ 𝑃𝑛 is total efficient if and only if 𝑚 is even and 𝑛 (mod (𝑚 + 1)) ∈ {1, 𝑚 − 2, 𝑚}. For example, the red vertices form a total efficient set in 𝑃6 □ 𝑃11 illustrated in Figure 9.8.

Figure 9.8 Total efficient set of red vertices in the grid 𝑃6 □ 𝑃11

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In 2008 Dejter [219] studied properties of perfect sets and total perfect sets in rectangular grids. In Figure 9.9, given in [219], the set of red vertices form a perfect set, since every vertex has exactly one red neighbor. This set of red vertices, however, is not an efficient set nor a total efficient set, since there are vertices having more than one red neighbor. For example, vertex (8, 4) has two red neighbors and vertex (2, 2) has three red neighbors.

Figure 9.9 Perfect set of red vertices in the grid 𝑃11 □ 𝑃16

9.3.3

Total Efficient Cylindrical Graphs

A cylindrical graph, or just a cylinder, is a Cartesian product of the form 𝑃𝑚 □ 𝐶𝑛 , or 𝐶𝑚 □ 𝑃𝑛 . The following three results on total efficient sets in cylinders 𝑃𝑚 □ 𝐶𝑛 are due to Kuziak et al. [549] in 2014. Theorem 9.31 ([549]) The cylinder 𝑃2 □ 𝐶𝑛 is total efficient when 𝑛 ≡ 0 (mod 3). Theorem 9.32 ([549]) The cylinder 𝑃2𝑚+1 □ 𝐶4𝑛 is total efficient for all 𝑚, 𝑛 ≥ 1. Let 𝑃𝑚 : 𝑢 0 𝑢 1 . . . 𝑢 𝑚−1 and 𝐶𝑛 : 𝑣 0 𝑣 1 . . . 𝑣 𝑛−1 𝑣 0 . For example, the set {𝑢 0 , 𝑢 4 , . . . , 𝑢 2𝑚−2 } × {𝑣 0 , 𝑣 1 , 𝑣 4 , 𝑣 5 , . . . , 𝑣 4𝑚−4 , 𝑣 4𝑚−3 } ∪ {𝑢 2 , 𝑢 6 , . . . , 𝑢 2𝑚 } × {𝑣 2 , 𝑣 3 , 𝑣 6 , 𝑣 7 , . . . , 𝑣 4𝑚−2 , 𝑣 4𝑚−1 } is a total efficient set of 𝑃2𝑚+1 □ 𝐶4𝑛 when 2𝑚 + 1 ≡ 3 (mod 4).

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Theorem 9.33 ([549]) For 3 ≤ 𝑛 ≤ 7, the following hold: (a) 𝑃𝑚 □ 𝐶3 is total efficient if and only if 𝑚 = 2. (b) 𝑃𝑚 □ 𝐶4 is total efficient if and only if 𝑚 is odd. (c) 𝑃𝑚 □ 𝐶5 is total efficient if and only if 𝑚 = 4. (d) 𝑃𝑚 □ 𝐶6 is total efficient if and only if 𝑚 = 2. (e) 𝑃𝑚 □ 𝐶7 is total efficient if and only if 𝑚 = 6.

9.3.4 Total Efficient Toroidal Graphs Some initial results on total efficient sets in toroidal graphs were given in 2008 by Dejter [219], who defined a total efficient set 𝑆 to be parallel if all edges in 𝐺 [𝑆] are parallel in orientation (either horizontal or vertical). Theorem 9.34 (Dejter’s Theorem [219]) A torus 𝐶𝑚 □ 𝐶𝑛 has a parallel total efficient set if and only if 𝑚, 𝑛 ≡ 0 (mod 4). An illustration of Theorem 9.34 is given in Figure 9.10.

Figure 9.10 A parallel total efficient set in the torus 𝐶4 □ 𝐶8 Dejter’s Theorem prompted Kuziak et al. [549] to make the following conjecture. Conjecture 9.35 ([549]) The torus 𝐶𝑚 □ 𝐶𝑛 is total efficient if and only if 𝑚, 𝑛 ≡ 0 (mod 4). The authors established partial results in support of their conjecture. The following result is based on the observation that if 𝐺 is an 𝑟-regular graph of order 𝑛 having a total efficient set, then 𝑛 ≡ 0 (mod 2𝑟). Theorem 9.36 ([549]) The torus 𝐶4 □ 𝐶𝑛 , for 𝑛 ≥ 4, is total efficient if and only if 𝑛 ≡ 0 (mod 4). Theorem 9.37 ([549]) The torus 𝐶𝑚 □ 𝐶𝑛 , for 𝑚 ≤ 𝑛 and 𝑚 ∈ {3, 5, 6, 7}, does not have a total efficient set. Proof We proceed with the following three claims. Claim 9.37.1 The torus 𝐶3 □ 𝐶𝑛 does not have a total efficient set for any 𝑛 ≥ 3.

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Proof Let 𝐺 = 𝐶3 □ 𝐶𝑛 where 𝑛 ≥ 3. Suppose, to the contrary, that 𝐺 has a total efficient set 𝑆. We note that in this case when 𝑚 = 3 that the vertices (1, 𝑗), (2, 𝑗), (3, 𝑗) induce a triangle in 𝐺 for all 𝑗 ∈ [𝑛]. No vertical edge (i.e., two adjacent vertices in 𝑆 that belong to the same triangle) can appear in any total efficient set since then the third vertex in that triangle is totally dominated twice. Hence, all edges that belong to 𝐺 [𝑆] must be horizontal, that is, between two adjacent vertices in 𝑆 in the same row. If 𝑛 = 3, then every horizontal edge also belongs to a triangle and, therefore, 𝑆 would contain no horizontal edge either, a contradiction. Hence, 𝑛 ≥ 4. Without loss of generality, we can assume by symmetry that the set 𝑆 contains the (horizontal) edge joining (1, 1) and (1, 2). Since every vertex is totally dominated exactly once by the set 𝑆, this means that no neighbor of (1, 1) different from (1, 2), and no neighbor of (1, 2) different from (1, 1), belongs to the set 𝑆. Thus, none of the vertices (2, 1), (3, 1), (2, 2), (3, 2), (1, 3), and (1, 𝑛) belongs to 𝑆. Further, since (2, 2) and (3, 2) are uniquely totally dominated by (1, 2), the vertices (2, 3) and (3, 3) do not belong to 𝑆. In order to totally dominate the vertices (2, 3) and (3, 3), the vertices (2, 4) and (3, 4) must belong to 𝑆, that is, the vertical edge joining (2, 4) and (3, 4) belongs to 𝑆. This is illustrated in Figure 9.11, where the vertices (1, 1), (1, 2), (2, 4), and (3, 4) are colored red. This, however, contradicts our earlier observation that no vertical edge belongs to 𝑆.

Figure 9.11 No total efficient set in the torus 𝐶3 □ 𝐶𝑛

Claim 9.37.2 The torus 𝐶5 □ 𝐶𝑛 does not have a total efficient set for any 𝑛 ≥ 5. Proof Let 𝐺 = 𝐶5 □ 𝐶𝑛 where 𝑛 ≥ 5. Suppose, to the contrary, that 𝐺 has a total efficient set 𝑆. Without loss of generality, we can assume that vertex (1, 1) belongs to the set 𝑆. Exactly one neighbor of every vertex must be in 𝑆. In particular, one neighbor of (1, 1) must be in 𝑆. We show that there must exist a vertical edge in 𝑆. Suppose, to the contrary, that every edge in 𝑆 is a horizontal edge. In particular, such a horizontal edge is incident with vertex (1, 1), that is, (1, 2) ∈ 𝑆 or (1, 𝑛) ∈ 𝑆, which by rotational symmetry are equivalent. So assume, without loss of generality, that vertex (1, 2) ∈ 𝑆. Since every vertex is totally dominated exactly once by the set 𝑆, this means that the vertices in the first two columns that belong to 𝑆 are the two vertices (1, 1) and (1, 2), that is, none of the vertices (2, 1), (3, 1), (4, 1), (5, 1), (2, 2), (3, 2), (4, 2), and (5, 2) belongs to 𝑆. From this it follows that the vertices (3, 3) and (4, 3) must be in 𝑆 in

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order to totally dominate the vertices (3, 2) and (4, 2), respectively. However, the edge incident with (3, 3) and (4, 3) is a vertical edge in 𝑆, a contradiction. Therefore, there must exist at least one vertical edge in 𝑆. Without loss of generality, we can assume that such a vertical edge is incident with vertex (1, 1), that is, (2, 1) ∈ 𝑆 or, by symmetry, (5, 1) ∈ 𝑆. Without loss of generality, we can assume that (2, 1) ∈ 𝑆. Since every vertex is totally dominated exactly once by the set 𝑆, this means that no neighbor of (1, 1) different from (2, 1) belongs to the set 𝑆, that is, none of the vertices (5, 1), (1, 2), and (1, 𝑛) belongs to 𝑆. Moreover, since no neighbor of (2, 1) different from (1, 1) belongs to the set 𝑆, none of the vertices (3, 1), (2, 2), and (2, 𝑛) belongs to 𝑆. Further, since (5, 1) is uniquely totally dominated by (1, 1), the vertices (4, 1), (5, 2), and (5, 𝑛) do not belong to 𝑆. Since (3, 1) is uniquely totally dominated by (2, 1), the vertices (3, 2) and (3, 𝑛) do not belong to 𝑆. We now consider vertex (4, 1), which as observed earlier cannot be in 𝑆. Thus, it must be dominated by either (4, 2) or (4, 𝑛), which by rotational symmetry are equivalent. So assume, without loss of generality, that vertex (4, 2) ∈ 𝑆. From this it follows that vertex (4, 3) must be in 𝑆 in order to totally dominate (4, 2). Since (4, 2) is the unique neighbor of (4, 3) that belongs to 𝑆, the vertices (3, 3), (5, 3), and (4, 4) do not belong to 𝑆. Consider next the two vertices (1, 3) and (2, 3) (see the two green vertices in Figure 9.12), which cannot be in 𝑆 and can only be totally dominated by the two red vertices (1, 4) and (2, 4) in Figure 9.12. Thus, both (1, 4) and (2, 4) belong to 𝑆. At this point, all vertices in the first four columns have been totally dominated exactly once. In particular, the vertices in the first four columns that belong to 𝑆 are the six red vertices in Figure 9.12, namely (1, 1), (2, 1), (4, 2), (4, 3), (1, 4), and (2, 4). Further, we note that neither (1, 5) nor (2, 5) belongs to 𝑆. At this point, if 𝑛 = 5, then no vertex in column 5 can be in 𝑆 and vertices (3, 5), (4, 5), and (5, 5) cannot be totally dominated, a contradiction. Thus, we can assume that 𝑛 ≥ 6. But in this case, vertices (3, 5), (4, 5), and (5, 5) must be totally dominated by the three vertices (3, 6), (4, 6), and (5, 6). However, then vertex (4, 6) has two neighbors in 𝑆, contradicting the fact that 𝑆 is a total efficient set.

Figure 9.12 No total efficient set in the torus 𝐶5 □ 𝐶𝑛 Claim 9.37.3 The torus 𝐶6 □ 𝐶𝑛 does not have a total efficient set for any 𝑛 ≥ 6.

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Proof Let 𝐺 = 𝐶6 □ 𝐶𝑛 , where 𝑛 ≥ 6. Suppose, to the contrary, that 𝐺 has a total efficient set 𝑆. Without loss of generality, we can assume that vertex (1, 1) belongs to the set 𝑆. We note that exactly one neighbor of (1, 1) must be in 𝑆. We show that there is no horizontal edge in 𝑆. Suppose, to the contrary, that such an edge exists. Without loss of generality, we can assume (1, 2) ∈ 𝑆, that is, the edge joining (1, 1) and (1, 2) is a horizontal edge. At this point, the only vertex in the first column that can be in 𝑆 different from (1, 1) is vertex (4, 1) and the only vertex in the second column that can be in 𝑆 different from (1, 2) is vertex (4, 2). Assume that (4, 1) ∈ 𝑆. It can only be totally dominated by (4, 𝑛) or (4, 2). Assume that it is totally dominated by vertex (4, 𝑛) in the rightmost column of the torus. Thus, (4, 2) is not in 𝑆, implying that vertex (3, 3) belongs to 𝑆 in order to totally dominate vertex (3, 2). Further, vertex (5, 3) belongs to 𝑆 in order to totally dominate vertex (5, 2). However, then vertex (4, 3) (see the green vertex (4, 3) in Figure 9.13) is totally dominated twice, a contradiction. Hence, (4, 𝑛) ∉ 𝑆 and so (4, 2) ∈ 𝑆 to totally dominate (4, 1). But then no vertex of column 3 is in 𝑆, that is, (𝑖, 3) ∉ 𝑆 for all 𝑖 ∈ [6]. This implies that (2, 4), (3, 4), (5, 4), and (6, 4) are in 𝑆 to totally dominate (2, 3), (3, 3), (5, 3), and (6, 3), respectively. But then each of (1, 4) and (4, 4) has two neighbors in 𝑆, contradicting the fact that 𝑆 is a total efficient set. Hence, (4, 1) ∉ 𝑆.

Figure 9.13 No total efficient set in the torus 𝐶6 □ 𝐶𝑛

Since (4, 1) ∉ 𝑆, the only vertex in the first column that belongs to 𝑆 is vertex (1, 1). Further, since neither (3, 2) nor (5, 2) is in 𝑆, the only vertex in 𝑆 that can totally dominate vertex (5, 1) is vertex (5, 𝑛) and the only vertex in 𝑆 that can totally dominate vertex (3, 1) is vertex (3, 𝑛). But then vertex (4, 𝑛) is totally dominated twice, a contradiction. Therefore, there is no horizontal edge in 𝑆. Recall that by our earlier assumption, (1, 1) ∈ 𝑆. Since every edge in 𝑆 is a vertical edge, we note that (2, 1) ∈ 𝑆 or (6, 1) ∈ 𝑆. By symmetry, we may assume (2, 1) ∈ 𝑆. Since exactly one neighbor of every vertex belongs to 𝑆, the vertices (1, 1) and (2, 1) are the only two vertices in the first column in 𝑆.

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In order to totally dominate vertex (4, 1), either (4, 2) or (4, 𝑛) belongs to 𝑆. In order to totally dominate vertex (5, 1), either (5, 2) or (5, 𝑛) belongs to 𝑆. Since every edge in 𝐺 [𝑆] is a vertical edge, there are two possible cases to consider, namely (4, 2) and (5, 2) are in 𝑆, or (4, 𝑛) and (5, 𝑛) are in 𝑆. If (4, 2) and (5, 2) belong to 𝑆, then vertex (3, 4) belongs to 𝑆 in order to totally dominate the vertex (3, 3). But then (1, 3) cannot be totally dominated by 𝑆 without creating a vertex which is totally dominated a second time. If (4, 𝑛) and (5, 𝑛) belong to 𝑆, then vertices (1, 𝑛 − 2) and (2, 𝑛 − 2) belong to 𝑆 in order to totally dominate the vertices (1, 𝑛 − 1) and (2, 𝑛 − 1), respectively. But then (3, 𝑛 − 1) cannot be totally dominated by 𝑆 without creating a vertex which is totally dominated a second time. Since both cases produce a contradiction. The proof of the case when 𝑚 = 7 is similar to the proofs we give for the cases when 𝑚 ∈ {3, 5, 6} but involves a slightly deeper case analysis. We therefore omit the details of the proof when 𝑚 = 7. The proof of Theorem 9.37 now follows from Claims 9.37.1, 9.37.2, and 9.37.3.

9.3.5

Total Efficient Product Graphs

In 2014 Kuziak et al. [549] studied efficient total domination in the lexicographic, strong, disjunctive, and Cartesian products of graphs. We mentioned in Sections 9.3.3 and 9.3.4 some of their results on Cartesian products, namely, cylinders and toroidal graphs. In this section, we present results for the other graph products. Recall that the lexicographic product 𝐺 ◦ 𝐻 of two graphs 𝐺 and 𝐻 is the graph with 𝑉 (𝐺 ◦ 𝐻) = 𝑉 (𝐺) × 𝑉 (𝐻), where two vertices (𝑢, 𝑣) and (𝑤, 𝑥) are adjacent whenever (i) (𝑢, 𝑤) ∈ 𝐸 (𝐺) or (ii) 𝑢 = 𝑤 and (𝑣, 𝑥) ∈ 𝐸 (𝐻). Notice that in a lexicographic product  𝐺 ◦ 𝐻, for any fixed vertex 𝑔 ∈ 𝑉 (𝐺), the set of vertices (𝑔, ℎ) : ℎ ∈ 𝑉 (𝐻) induces a subgraph of 𝐺 ◦ 𝐻 that is isomorphic to 𝐻; we call this an 𝐻-layer of 𝐺 ◦ 𝐻. Theorem 9.38 ([549]) The lexicographic product 𝐺 ◦ 𝐻 of two graphs 𝐺 and 𝐻 has a total efficient set if and only if either (a) 𝐺 is an empty graph and 𝐻 has a total efficient set, or (b) 𝐺 has a total efficient set and 𝐻 contains an isolated vertex. Proof If 𝐺 is an empty graph of order 𝑛, then 𝐺 ◦ 𝐻 is isomorphic to 𝑛 disjoint copies of 𝐻. If 𝐻 has a total efficient set, then 𝑛 copies of 𝐻 is a graph having a total efficient set. Thus, if (a) holds, then 𝐺 ◦ 𝐻 has a total efficient set. Suppose that (b) holds, that is, 𝐺 has a total efficient set 𝐷 𝐺 and there is an isolated vertex 𝑣 0 in 𝐻. We note that in 𝐺 ◦ 𝐻, we have N𝐺◦𝐻 ((𝑔, 𝑣 0 )) = N𝐺 (𝑔) × 𝑉 (𝐻) and Ø N((𝑔, 𝑣 0 )) = 𝑉 (𝐺 × 𝐻). 𝑔∈𝐷𝐺

If 𝑔, 𝑔 ′ ∈ 𝐷 𝐺 and 𝑔 ≠ 𝑔 ′ , then N𝐺◦𝐻 ((𝑔, 𝑣 0 )) ∩ N𝐺◦𝐻 ((𝑔 ′ , 𝑣 0 )) ≠ ∅ implies that N𝐺 (𝑔) ∩ N𝐺 (𝑔 ′ ) ≠ ∅, which is a contradiction. Therefore, 𝐷 𝐺 × {𝑣 0 } is a total efficient set of 𝐺 ◦ 𝐻. Thus, if (b) holds, then 𝐺 ◦ 𝐻 has a total efficient set.

Section 9.3. Efficient Total Domination

281

Conversely, let 𝐺 ◦ 𝐻 have a total efficient set 𝐷 and let (𝑔, ℎ) and (𝑔 ′ , ℎ′ ) be adjacent vertices in 𝐷. Suppose first that there exists an edge with 𝑔 ≠ 𝑔 ′ . If ℎ′′ ∈ N 𝐻 (ℎ), then (𝑔, ℎ′′ ) ∈ N𝐺◦𝐻 ((𝑔, ℎ)) ∩ N𝐺◦𝐻 ((𝑔 ′ , ℎ′ )), which is a contradiction. Hence, ℎ (and by symmetry also ℎ′ ) is an isolated vertex of 𝐻. Since 𝐻 contains an isolated vertex, it follows that 𝐺 has no isolated vertices, otherwise 𝐺 ◦ 𝐻 would contain isolated vertices, which is impossible for a graph having a total efficient set. Thus, if (𝑔, ℎ) and (𝑔 ′ , ℎ′ ) are adjacent vertices in 𝐷, then ℎ and ℎ′ are isolated vertices of 𝐻 (notice that it can happen that ℎ = ℎ′ ). If (𝑔, ℎ) ∈ 𝑉 (𝐺 ◦ 𝐻), let p𝐺 ((𝑔, ℎ)) = {𝑔}, the projection of (𝑔, ℎ) onto the vertex 𝑔 in the graph 𝐺. Similarly, for a total efficient set 𝐷 of 𝐺 ◦ 𝐻, let p𝐺 (𝐷) denote the set of vertices that are projections of vertices in 𝐷 onto the vertices of the graph 𝐺. If 𝑔 ′′ ∈ N𝐺 (𝑔) ∩N𝐺 (𝑔 ′ ) for some 𝑔, 𝑔 ′ ∈ p𝐺 (𝐷), then 𝑔 ′′ × 𝐻 ⊆ N𝐺◦𝐻 ((𝑔, ℎ)) ∩ N𝐺◦𝐻 ((𝑔 ′ , ℎ′ )) for (𝑔, ℎ), (𝑔 ′ , ℎ′ ) ∈ 𝐷, which is a contradiction. In addition, Ø N𝐺 (𝑔 ′′ ) = 𝑉 (𝐺), 𝑔′′ ∈p𝐺 (𝐷)

since Ø

N𝐺◦𝐻 ((𝑔1 , ℎ1 )) (𝑔1 ,ℎ1 ) ∈𝐷

= 𝑉 (𝐺 ◦ 𝐻),

and 𝐷 is a total efficient set of 𝐺 ◦ 𝐻. Thus, 𝐺 has a total efficient set p𝐺 (𝐷). Now we can assume that adjacent vertices in 𝐷 have the same first coordinate, that is, any edge between two vertices in 𝐷 has the form (𝑔, ℎ) (𝑔, ℎ′ ). Thus, 𝑔 is an isolated vertex of 𝐺, otherwise {𝑔 ′ } × 𝐻 ⊆ N𝐺◦𝐻 ((𝑔, ℎ)) ∩ N𝐺◦𝐻 ((𝑔, ℎ′ )) for any neighbor 𝑔 ′ of 𝑔 in 𝐺, which is not possible. Since N𝐺◦𝐻 ((𝑔, ℎ)) : (𝑔, ℎ) ∈ 𝐷 forms a partition of 𝑉 (𝐺 ◦ 𝐻), every vertex (𝑔 ′′ , ℎ′′ ) is in some N𝐺◦𝐻 ((𝑔, ℎ)). Again, (𝑔 ′′ , ℎ′′ ) is in some N𝐺◦𝐻 ((𝑔 ′ , ℎ′ )) and we have 𝑔 = 𝑔 ′ = 𝑔 ′′ . Hence, every vertex of 𝐺 is an isolated vertex. Assume that 𝐺 has order 𝑛. Every 𝐻-layer is isomorphic to 𝐻 and 𝐺 ◦ 𝐻 is isomorphic to 𝑛 disjoint copies of 𝐻. Since 𝐺 ◦ 𝐻 has a total efficient set, every component of 𝐺 ◦ 𝐻 has a total efficient set. Therefore, 𝐻 has a total efficient set. The strong product 𝐺 ⊠ 𝐻 of two graphs 𝐺 and 𝐻 is a graph with 𝑉 (𝐺 ⊠ 𝐻) = 𝑉 (𝐺) × 𝑉 (𝐻), where two vertices (𝑢, 𝑣) and (𝑤, 𝑥) are adjacent whenever (i) 𝑢𝑤 ∈ 𝐸 (𝐺) and 𝑣 = 𝑥, or (ii) 𝑢 = 𝑤 and 𝑣𝑥 ∈ 𝐸 (𝐻), or (iii) 𝑢𝑤 ∈ 𝐸 (𝐺) and 𝑣𝑥 ∈ 𝐸 (𝐻). Note that in any strong product 𝐺 ⊠ 𝐻, we have |N(𝑢, 𝑤) ∩ N(𝑣, 𝑥)| ≥ 2 for any two adjacent vertices (𝑢, 𝑣) and (𝑤, 𝑥), where both vertices are not isolated vertices of 𝐺 and 𝐻, respectively. Thus, the following becomes clear. Theorem 9.39 ([549]) The strong product 𝐺 ⊠ 𝐻 of two graphs has a total efficient set if and only if one factor is the empty graph and the other graph has a total efficient set. The disjunctive product 𝐺 ⊕ 𝐻 is a graph with 𝑉 (𝐺 ⊕ 𝐻) = 𝑉 (𝐺) × 𝑉 (𝐻), where two vertices (𝑢, 𝑣) and (𝑤, 𝑥) are adjacent whenever 𝑢𝑤 ∈ 𝐸 (𝐺) or 𝑣𝑥 ∈ 𝐸 (𝐻).

282

Chapter 9. Efficient Domination in Graphs

Theorem 9.40 ([549]) The disjunctive product 𝐺 ⊕ 𝐻 of two graphs 𝐺 and 𝐻 has a total efficient set if and only if one graph has a total efficient set and the other graph contains an isolated vertex. In 2008 Abay-Asmerom et al. [1] added the following result about total efficient sets in direct products (also called tensor products). The direct product 𝐺 × 𝐻 is a graph with 𝑉 (𝐺 × 𝐻) = 𝑉 (𝐺) × 𝑉 (𝐻), where two vertices (𝑢, 𝑣) and (𝑤, 𝑥) are adjacent whenever 𝑢𝑤 ∈ 𝐸 (𝐺) and 𝑣𝑥 ∈ 𝐸 (𝐻). Notice that in the direct product 𝐺 × 𝐻, N((𝑢, 𝑤)) = N𝐺 (𝑢) × N 𝐻 (𝑤). Theorem 9.41 ([1]) The direct product 𝐺 × 𝐻 of two graphs 𝐺 and 𝐻 has a total efficient set if and only if both 𝐺 and 𝐻 have a total efficient set. Corollary 9.42 ([1]) If graphs 𝐺 and 𝐻 both have a total efficient set, then 𝛾t (𝐺 × 𝐻) = 𝛾t (𝐺)𝛾t (𝐻).

9.3.6

Total Efficient Circulant Graphs

In 2014 Castle et al. [139] characterized the graphs in two subclasses of Cayley graphs which have total efficient sets, thereby extending the results in 2002 by Gavlas and Schultz [328], who characterized which cycles have total efficient sets. In 2017 Feng et al. [291] characterized the total efficient 𝑝-regular Cayley graphs, where 𝑝 is an odd prime, as follows. Theorem 9.43 ([291]) For 𝑛 ≥ 1 and 𝑝 an odd prime, a connected 𝑝-regular circulant graph Cay(Z𝑛 , 𝑆) has a total efficient set if and only if 𝑝 divides 𝑛 and 𝑠 . 𝑠′ (mod 𝑝), for distinct 𝑠, 𝑠′ ∈ 𝑆 ∪ {0}. In 2020 Kwon et al. [551] gave necessary and sufficient conditions for the existence of total efficient sets in 4-regular circulant graphs. Note that if a 4-regular circulant graph has a total efficient set, then its order must be a multiple of 8. Theorem 9.44 ([551]) A connected circulant graph 𝐶8𝑚 ⟨𝑎, 𝑏⟩ has a total efficient set if and only if (a) 𝑏 ≡ 3 (mod 8) or

(b) 𝑏 ≡ 1 (mod 8) and gcd 8𝑚, |𝑎 − 𝑏| = gcd 4𝑚, |𝑎 − 𝑏| . One can observe from this theorem that any connected 4-regular circulant graph of the form 𝐶8𝑚 ⟨𝑎, 𝑏⟩ is isomorphic to a circulant graph of the form 𝐶8𝑚 ⟨𝑐, 𝑑⟩, where 𝑐 ≡ 1 (mod 8) and 𝑑 (mod 8) ∈ [4] 0 . We note in closing this section that also in 2020 Kwon and Sohn [550] characterized 5-regular circulant graphs 𝐶10𝑚 ⟨𝑎, 𝑏, 5𝑚⟩ having total efficient sets.

9.4

Algorithms and Complexity of Efficient Domination

The most common decision problems corresponding to efficient domination are the following.

Section 9.4. Algorithms and Complexity of Efficient Domination

283

EFFICIENT DOMSET (ED)

Instance: Graph 𝐺 = (𝑉, 𝐸) Question: Does 𝐺 have an efficient dominating set? EFFICIENT TOTAL DOMSET (ETD)

Instance: Graph 𝐺 = (𝑉, 𝐸) Question: Does 𝐺 have an efficient total dominating set? PERFECT DOMSET (PD)

Instance: A graph 𝐺 = (𝑉, 𝐸) and a positive integer 𝑘. Question: Does 𝐺 have a perfect dominating set of cardinality at most 𝑘? PERFECT TOTAL DOMSET (PTD)

Instance: A graph 𝐺 = (𝑉, 𝐸) and a positive integer 𝑘. Question: Does 𝐺 have a perfect total dominating set of cardinality at most 𝑘? There are also weighted versions of these decision problems, where weights (positive integers) are assigned to the vertices. If a graph 𝐺 has an efficient dominating set, efficient total dominating set, or perfect dominating set, then one seeks to minimize the sum of the weights of the vertices in such a set. The weighted versions of these efficient domination problems are denoted WED, WETD, and WPD, respectively. In this section, we review only a few of the many NP-completeness results and algorithms for special classes of graphs. Bange et al. [56] showed that ED is NP-complete for arbitrary graphs, using a straightforward transformation from 3SAT. They constructively characterized trees that have an efficient dominating set and showed how to determine in linear time the maximum number of vertices in a tree 𝑇 that can be efficiently dominated. They also presented a recursive characterization of trees having two disjoint efficient dominating sets. In 1991 Fellows and Hoover [290] studied efficient domination, efficient total domination, and perfect domination and showed that all three corresponding decision problems are NP-complete for planar graphs, even for maximum degree at most 3, while the efficient domination problem ED can be solved in linear time for trees. In particular, for efficient domination they proved the following. Theorem 9.45 ([290]) EFFICIENT DOMSET (ED) is NP-complete for planar graphs having maximum degree 3. Proof The ED problem is clearly in the class NP since it is easy to guess a possible solution 𝑆 and verify if 𝑆 is an efficient dominating set in polynomial time.

284

Chapter 9. Efficient Domination in Graphs

To show that ED is NP-complete, one can construct a transformation from the following NP-complete problem to ED. THREE-DIMENSIONAL MATCHING (3DM)

Instance: Three disjoint sets 𝑅, 𝐵, and 𝑌 of equal cardinality 𝑞 and a set 𝑇 of triples, 𝑇 ⊆ 𝑅 × 𝐵 × 𝑌 , where each element of 𝑅 ∪ 𝐵 ∪ 𝑌 belongs to at least one triple in 𝑇. Question: Does there exist a set of 𝑞 triples in 𝑇 that contains all elements of 𝑅 ∪ 𝐵 ∪ 𝑌? Associated with each instance 𝐼 = (𝑇, 𝑅, 𝐵, 𝑌 ) of 3DM, one can construct a bipartite graph 𝐺 𝐼 = (𝑇, 𝑅 ∪ 𝐵 ∪ 𝑌 , 𝐸) with partite sets 𝑇 and 𝑅 ∪ 𝐵 ∪ 𝑌 , where a vertex 𝑥 ∈ 𝑅 ∪ 𝐵 ∪ 𝑌 is adjacent to a vertex 𝑡 ∈ 𝑇 if and only if the triple 𝑡 contains 𝑥. To the graph 𝐺 𝐼 , add a copy 𝑇 ′ of the set 𝑇 and the edges of a perfect matching between the vertices of 𝑇 and the vertices of 𝑇 ′ , whereby a vertex 𝑡 ∈ 𝑇 is joined to its copy 𝑡 ′ ∈ 𝑇 ′ . Thus, every vertex in 𝑇 ′ becomes a leaf. Denote the resulting bipartite graph by 𝐺 ′𝐼 . If 𝑃 ⊆ 𝑇 is a solution to the instance 𝐼 = (𝑇, 𝑅, 𝐵, 𝑌 ) of 3DM, then 𝑃 ∪ {𝑡 ′ : 𝑡 ∈ 𝑇 − 𝑃} is an efficient dominating set of 𝐺 ′𝐼 . Conversely, if 𝐺 ′𝐼 has an efficient dominating set 𝑆, then no vertex in 𝑅 ∪ 𝐵 ∪ 𝑌 can be in 𝑆, for if a vertex 𝑥 ∈ 𝑅 ∪ 𝐵 ∪ 𝑌 is in 𝑆, then there is at least one triple 𝑡 ∈ 𝑇 to which 𝑥 belongs. This means that 𝑡 ∉ 𝑆. But now vertex 𝑡 ′ must be in 𝑆 or else it is not dominated; but if 𝑡 ′ ∈ 𝑆 then vertex 𝑡 is dominated twice and 𝑆 is not an efficient dominating set. Thus, any efficient dominating set 𝑆 in 𝐺 ′𝐼 satisfies 𝑆 ⊆ 𝑇 ∪ 𝑇 ′ . This must constitute a solution set for 𝐼 since each element of 𝑅 ∪ 𝐵 ∪ 𝑌 must have exactly one neighbor in 𝑆, that is, one triple to which it belongs. In order to complete the proof of the theorem, Fellows and Hoover [290] note that Dyer and Frieze showed in [253] that 3DM is NP-complete even when restricted to instances for which the associated bipartite graph 𝐺 𝐼 is planar. Thus, given an instance of 3DM in which the bipartite graph 𝐺 ′𝐼 is planar, one can, in polynomial time, transform 𝐺 ′𝐼 to a planar bipartite graph 𝐺 ′′𝐼 having maximum degree at most 3 such that 𝐺 ′𝐼 has an efficient dominating set if and only if the revised graph 𝐺 ′′𝐼 has an efficient dominating set. This transformation can be described as follows. Let 𝑆1,1,3 be a tree obtained from a star 𝐾1,3 by subdividing one edge twice; thus, it becomes a spider having a central vertex 𝑤 𝑖 of degree 3 to which are attached two leaves, labeled 𝑥𝑖 and 𝑦 𝑖 and a path of length 3, ending in a leaf labeled 𝑧𝑖 . Suppose, for example, that the graph 𝐺 ′𝐼 has a vertex 𝑣 ∈ 𝑅 ∪ 𝐵 ∪ 𝑌 of degree 8. Label its neighbors 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓 , 𝑔, ℎ. Create a path 𝑃 : 𝑟 0 𝑟 1 𝑟 2 . . . . Let 𝑟 0 = 𝑎. Attach a spider 𝑆1,1,3 to vertex 𝑟 1 , so 𝑟 1 = 𝑧1 , and identify 𝑥1 = 𝑏 and 𝑦 1 = 𝑐. Attach a second spider to vertex 𝑟 4 = 𝑧 2 , and identify 𝑥 2 = 𝑑 and 𝑦 2 = 𝑒. Attach a third spider to vertex 𝑟 7 = 𝑧3 and identify 𝑥3 = 𝑓 and 𝑦 3 = 𝑔. Finally, let 𝑟 11 = ℎ. In this way each attached spider has two leaves which correspond to two of the neighbors of

Section 9.4. Algorithms and Complexity of Efficient Domination

285

the original vertex 𝑣. The first neighbor of vertex 𝑣 is the initial vertex of this path 𝑟 0 = 𝑎, and if the degree of 𝑣 is even, the path 𝑃 will have an ending path, attached to the last spider vertex 𝑧 𝑘 , of length 4, whose leaf is the final neighbor of the vertex 𝑣.
Fellows and
Hoover further showed that ED in trees can be solved in O ln(|𝑉 |) time with O |𝑉 | processors in the CREW PRAM (Concurrent Read Exclusive Write,
Parallel Random Access Model) model of parallel computation, and also in O |𝑉 | time sequentially. In her 1994 PhD thesis, McRae [588] obtained the following eight NP-completeness results, none of which have been published. We present two of her proofs here. Theorem 9.46 ([588]) ED is NP-complete, even when restricted to the class of chordal graphs, the class of line graphs, or the class of line graphs of bipartite graphs. Theorem 9.47 ([588]) ETD is NP-complete, even when restricted to the class of line graphs, or the class of line graphs of bipartite graphs. Theorem 9.48 ([588]) PD is NP-complete, even when restricted to the class of line graphs, or the class of line graphs of bipartite graphs. Theorem 9.49 ([588]) Unless P = NP, there is no polynomial algorithm that can take as input a bipartite graph 𝐺 and find a perfect dominating set for 𝐺 with cardinality that is within a factor 𝑓 of a minimum perfect dominating set. McRae’s NP-completeness results are based on simple transformations from the following well-known NP-complete problem. EXACT COVER BY 3-SETS (X3C)

Instance: Set 𝑋 = {𝑥 1 , 𝑥2 , . . . , 𝑥 3𝑞 } of elements, with |𝑋 | = 3𝑞, a collection 𝐶 = {𝐶1 , 𝐶2 , . . . , 𝐶𝑚 } of 3-element subsets of 𝑋. Question: Does 𝐶 contain an exact cover for 𝑋, that is, a subcollection 𝐶 ′ ⊆ 𝐶 such that every element of 𝑋 occurs in exactly one subset in 𝐶 ′ ? Theorem 9.50 ([588]) ED is NP-complete for bipartite graphs and for chordal graphs. Proof It is easy to see that ED is in NP since one can verify in polynomial time whether an arbitrary set 𝑆 is an efficient dominating set. We construct a polynomial time transformation from X3C to ED as follows. Given an instance (𝑋, 𝐶) of X3C, we construct the following instance 𝐺 of ED. For each element 𝑥𝑖 ∈ 𝑋, construct a vertex 𝑥 𝑖 ∈ 𝑉 (𝐺). For each subset 𝐶𝑖 = {𝑥 𝑖1 , 𝑥𝑖2 , 𝑥𝑖3 } ∈ 𝐶, construct a copy of 𝑃2 with two adjacent vertices 𝑐 𝑖1 and 𝑐 𝑖2 . From each vertex 𝑐 𝑖1 , add three edges to the three vertices in the subset 𝐶𝑖 . Note that this graph 𝐺 is bipartite. One can show that 𝐺 has an efficient dominating set if and only the instance of X3C has a solution. Notice that if 𝐺 has an efficient dominating set 𝑆, then 𝑆

286

Chapter 9. Efficient Domination in Graphs

cannot contain any of the 𝑥 𝑗 vertices, else some of the 𝑐 𝑖2 vertices cannot be efficiently dominated. Notice, as well, that if a 𝑐 𝑖1 vertex is not in 𝑆, then 𝑐 𝑖2 must be in 𝑆. If the graph 𝐺 is modified so that all possible edges between 𝑥 𝑗 vertices are added to form a complete subgraph of order 𝑛, then the resulting graph 𝐺 ′ is chordal. The same argument can then be used to show that 𝐺 ′ has an efficient dominating set if and only if the instance of X3C has a solution. Thus, ED is NP-complete for bipartite graphs and for chordal graphs. Theorem 9.51 ([588]) ETD is NP-complete for bipartite graphs and for chordal graphs. Proof It is easy to see that ETD is in NP since one can verify in polynomial time whether an arbitrary set 𝑆 is an efficient total dominating set. We construct a polynomial time transformation from X3C to ETD as follows. Given an instance (𝑋, 𝐶) of X3C, we construct the following instance 𝐺 of ETD. For each element 𝑥 𝑖 ∈ 𝑋, construct a vertex 𝑥𝑖 ∈ 𝑉 (𝐺). For each subset 𝐶𝑖 = {𝑥𝑖1 , 𝑥𝑖2 , 𝑥𝑖3 } ∈ 𝐶, construct a copy of 𝑃3 with vertices 𝑐 𝑖1 , 𝑐 𝑖2 , and 𝑐 𝑖3 . From each vertex 𝑐 𝑖1 , add three edges to the three vertices in the subset 𝐶𝑖 . Note that this graph 𝐺 is bipartite. One can show that 𝐺 has an efficient total dominating set if and only if the instance of X3C has a solution. Notice that if 𝐺 has an efficient total dominating set 𝑆, then 𝑆 cannot contain any of the 𝑥 𝑗 vertices, else some of the 𝑐 𝑖2 and 𝑐 𝑖3 vertices cannot be efficiently total dominated. Notice, as well, that if a 𝑐 𝑖1 vertex is in 𝑆 in order to efficiently dominate three 𝑥 𝑗 vertices, then 𝑐 𝑖2 must also be in 𝑆. Similarly, if 𝑐 𝑖1 is not in 𝑆, then 𝑐 𝑖2 and 𝑐 𝑖3 must be in 𝑆. If the graph 𝐺 is modified so that all possible edges between 𝑥 𝑗 vertices are added to form a complete subgraph of order 𝑛, then the resulting graph 𝐺 ′ is chordal. The same argument can then be used to show that 𝐺 ′ has an efficient total dominating set if and only if the instance of X3C has a solution. Thus, ETD is NP-complete for bipartite graphs and for chordal graphs. In 2002 Lu and Tang [574] showed that ED is NP-complete for planar bipartite graphs and chordal bipartite graphs. In 1996 Yen and Lee [763] were among the first to study perfect domination in graphs. The authors studied the following three variants of perfect dominating sets 𝑆: • 𝑆 is an independent perfect dominating set if 𝑆 is an independent set. Note that an independent perfect dominating set is equivalent to an efficient dominating set. • 𝑆 is a total perfect dominating set if the subgraph 𝐺 [𝑆] induced by 𝑆 is isolate-free. • 𝑆 is a connected perfect dominating set if the subgraph 𝐺 [𝑆] induced by 𝑆 is connected. Yen and Lee [763] showed that these three variants of perfect domination are NPcomplete for bipartite graphs, and independent perfect domination and total perfect domination are NP-complete for chordal graphs. But for the family of block graphs, the authors constructed linear algorithms for weighted perfect domination and each of these three variants of perfect domination. Tables 9.2, 9.3, and 9.4 give a brief summary of algorithm and complexity results for efficient domination and the several variations of efficient domination discussed

Section 9.4. Algorithms and Complexity of Efficient Domination

287

in this chapter. We also refer the reader to the 2018 survey by Brandstädt [94]. Let the graphs in the table have order 𝑛, size 𝑚, minimum degree 𝛿, and maximum degree Δ.

Problem

Family

Author(s)

Year

Ref

ED

general cubic planar 3-regular planar bipartite, chordal general

Bange et al. Kratochvíl and Křivánek Livingston and Stout Yen and Lee Bakker and Van Leeuwen Fellows and Hoover McRae

1988 1988 1990 1990 1991

[56] [541] [568] [761] [48]

1991 1994

[290] [588]

McRae

1994

[588]

Kratochvíl et al. Smart and Slater

1995 1995

[542] [681]

Yen and Lee Yen and Lee Bange et al. Lu and Tang

1996 1996 1996 1998

[763] [763] [54] [573]

Lu and Tang

2002

[574]

Schaudt Eschen and Wang Brandstädt et al. Abrishami and Rahbarnia Brandstädt and Mosca

2012 2014 2018 2018

[668] [262] [98] [6]

2020

[108]

PD, ED ED PD ETD ED ED, ETD

PD

ETD ED ED TPD ED ED ED ETD ED WED ED ED

planar, Δ ≤ 3 bipartite, chordal line graphs, line graphs of bipartite line graph, line graphs of bipartite, NP-approximation chordal chordal, 2𝑃3 -free chordal chordal bipartite, chordal general claw-free, line graphs of bipartite chordal bipartite, planar bipartite planar bipartite, Δ ≤ 3 chordal 2𝑃3 -free diam = 3𝑘 or 3𝑘 + 2 𝐾1,5 -free chordal, 𝐾3 + 3𝐾1 -free chordal

Table 9.2 NP-completeness results for efficient domination

Chapter 9. Efficient Domination in Graphs

288 Problem ED WPD WED WED WED WPD WED ETD WPD WED WED

WED WED WED WED ED WED

ED WED WETD

ETD WED WED

Family

Author(s)

Bange et al. trees trees Yen and Lee Chang and Liu split graphs circular-arc Chang and Liu interval Chang and Liu cocomparability Chang et al. interval Chang et al. interval Kratochvíl et al. series-parallel Yen and Lee block graphs Yen and Lee trapezoid Liang et al. graphs permutation cocomparability Chang trapezoid Lin circular-arc Chang Courcelle et al. bounded clique-width distance-hereditary Hsieh bipartite Lu and Tang permutation distancehereditary simplicial graphs Barbosa and Slater convex bipartite Brandstädt et al. interval bigraphs chordal bipartite, Schaudt balanced, odd-sun-free chordal, strongly chordal 𝑇3 -free chordal, Schaudt claw-free (𝑆1,2,2 , Net)-free Milanič 𝑃5 -free Brandstädt et al.

Year Ref

Time

1988 [56] linear 1990 [761] linear 1993 [148] linear 1994 [149] O (𝑛2 + 𝑚) 1994 [149] O (𝑛 + 𝑚) 1995 [145] O (𝑛𝑚) 1995 [145] O (𝑛 + 𝑚) 1995 [542] O (𝑛3 ) 1995 [762] linear 1996 [763] O (𝑛 + 𝑚)
1997 [562] O★ 𝑛 ln ln(𝑛)
+𝑚 O (𝑛 + 𝑚) 1997 [146] O (𝑛2 ) 1998 [565] O★ (𝑛 ln(𝑛)) 1998 [147] linear 2000 [203] polynomial 2002 [501] O (𝑛 + 𝑚) 2002 [574] O★ (𝑛) O★ (𝑛) 2012 [59] 2012 [103]

polynomial polynomial

2012 [668]

polynomial

2012 [668]

O (𝑛3 )

2013 [591] 2013 [104]

polynomial polynomial

Table 9.3 Polynomial results for efficient domination

Section 9.4. Algorithms and Complexity of Efficient Domination Problem

Family

Author(s)

Brandstädt and Le (𝑆1,2,2 , xNet)-free (𝑃5 ∪ 𝑘 𝑃2 )-free Brandstädt and Giakoumakis WED 𝑃5 -free Brandstädt AT-free Brandstädt et al. WED dually chordal ED (𝑃6 , bull)-free Brandstädt and Karthick (𝑃6 , 𝑆1,1,3 )-free WED 𝑃6 -free Lokshtanov et al. WED 𝑃5 -free, Brandstädt and Mosca 𝑃6 -free WED (𝑃6 , banner)-free Karthick WED net-free chordal, Brandstädt and Mosca 𝑆1,2,3 -free chordal WED (𝑃6 , house, hole, Brandstädt et al. domino)-free 𝑃6 -free chordal (𝑃6 , net)-free WED diam-3 bipartite Abrishami and Rahbarnia diam-3 planar WED 2𝑃2 -free Brandstädt et al. (𝑃4 ∪ 𝑃2 )-free (2𝑃3 , 𝑘 𝑃2 )-free connected (𝐾3 , 𝑆1,2,3 )-free (2𝑃3 , 𝐾3 )-free WED 𝑃6 -free Lokshtanov et al. Brandstädt and Mosca WED 𝐻-free bipartite for 𝐻 = 𝑃7 or 𝐻 = 𝑃9 , Δ ≤ 3 or 𝐻 = 𝑆2,2,4 WED 𝑆1,2,3 -free chordal Brandstädt and Mosca net-free chordal extended-gem-free chordal WED 𝐻-free chordal, Brandstädt and Mosca for 𝐻 chordal with |𝑉 (𝐻)| ≤ 5 with four exceptions WED WED

289

Year Ref 2014 2014 2015 2015 2016 2016 2016 2016 2017

Time

[101] polynomial [97] polynomial [93] linear [96] polynomial O (𝑚 + 𝑛) [99] polynomial polynomial [570] polynomial [105] linear O (𝑛5 𝑚) [517] O (𝑛3 ) [106] polynomial

2017 [95] polynomial polynomial O (𝑛2 𝑚) 2018 [6] O (𝑛 + 𝑚) O (𝑛5 ) 2018 [98] linear O (𝛿𝑚) polynomial polynomial O (𝑛3 𝑚) 2018 [571] polynomial 2019 [107] polynomial polynomial O (𝑛6 ) 2020 [108] polynomial polynomial polynomial 2020 [108] polynomial

Table 9.4 Polynomial results for efficient domination

Chapter 10

Domination and Forbidden Subgraphs 10.1 Introduction In this chapter, we study the three core domination parameters in graphs with specific structural restrictions imposed, such as forbidding certain cycles or claws. First we review some terminology. A graph 𝐺 is 𝐹-free if it does not contain a given graph 𝐹 as an induced subgraph. Thus, if 𝐺 is 𝐶 𝑘 -free for an integer 𝑘 ≥ 3, then 𝐺 contains no induced cycle 𝐶 𝑘 and we say that 𝐺 has no induced 𝑘-cycle. In the special case when 𝐹 = 𝐾1,3 , an 𝐹-free graph is called claw-free, while if 𝐹 = 𝐾4 − 𝑒, then an 𝐹-free graph is called diamond-free. Similarly, a 𝐾3 -free graph is called triangle-free. A graph 𝐺 is (𝐹1 , 𝐹2 , . . . , 𝐹𝑘 )-free, for a collection of graphs 𝐹1 , 𝐹2 , . . . , 𝐹𝑘 , if 𝐺 contains no induced 𝐹𝑖 for any 𝑖 ∈ [𝑘]. For example, a (𝐶4 , 𝐶5 )-free graph has no induced 4-cycle and no induced 5-cycle. In the more general case where the restriction forbids a subgraph 𝐻 that is not necessarily induced, we simply say that 𝐺 does not contain 𝐻 as a subgraph, or that 𝐺 has no 𝐻 subgraph.

10.2

Domination and Forbidden Cycles

In this section, we show that if certain cycles are forbidden, then the upper bounds on the three core domination parameters established in Chapter 6 can often be improved. Hence, the lack of certain cycles in a sense decreases these domination numbers.

10.2.1

Domination Number

We present improved bounds on the domination number of a graph when certain cycles are forbidden. © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_10

291

292

Chapter 10. Domination and Forbidden Subgraphs

Forbidden 4- and 5-Cycles and Minimum Degree Two Recall that in Chapter 6, we defined the family Gdom to consist of all graphs that can be obtained from a graph in the family Fdom by adding edges, including the possibility of none, joining link vertices. For example, a graph 𝐺 in the family Gdom constructed from a graph in the family Fdom with two cycle units and two key units is shown in Figure 10.1, where the highlighted vertices are link vertices.

Figure 10.1 A graph in the family Gdom As shown in Theorem 6.19, if 𝐺 is a connected graph of order 𝑛 > 10 with 𝛿(𝐺) ≥ 2, then the extremal graphs 𝐺 achieving the upper bound of 𝛾(𝐺) = 25 𝑛 for the domination number, are precisely the graphs in the family Gdom . We note that every graph in the family Gdom contains a 4-cycle or a 5-cycle. Hence, it is a natural question to ask if the 25 -bound on the domination number can be improved if we forbid 4- and 5-cycles, or if this bound is still asymptotically best possible. In 2011 Henning et al. [477] showed that the 25 -bound can indeed be significantly improved by forbidding 4- and 5-cycles. In fact, they proved a much stronger result, which requires some additional terminology. Definition 10.1 A vertex 𝑣 is called a bad-4-cut-vertex of a graph 𝐺 if 𝐺 −𝑣 contains a component 𝐶𝑣 , which is an induced 4-cycle, and 𝑣 is adjacent to at least one but at most three vertices on 𝐶𝑣 . Let bc4 (𝐺) denote the number of bad-4-cut-vertices in 𝐺. Definition 10.2 A cycle 𝐶 is called a special-cycle if 𝐶 is a 5-cycle in 𝐺 such that if 𝑢 and 𝑣 are consecutive vertices on 𝐶, then at least one of 𝑢 and 𝑣 has degree 2 in 𝐺. Let sc(𝐺) be the maximum number of vertex-disjoint special-cycles in 𝐺 that contain no bad-4-cut-vertices. As observed in [477], if 𝐺 is a graph of order 𝑛, then sc(𝐺) + bc4 (𝐺) ≤ 15 𝑛. By contracting two vertices 𝑥 and 𝑦 in a graph 𝐺, we mean replacing the vertices 𝑥 and 𝑦 by a new vertex 𝑣 𝑥 𝑦 and joining 𝑣 𝑥 𝑦 to all vertices in 𝑉 \ {𝑥, 𝑦} that were adjacent to 𝑥 or 𝑦 in 𝐺. The authors in [477] defined two types of reducible graphs. Using these reductions, they defined a family F≤13 of graphs of order at most 13. Definition 10.3 If a graph 𝐺 has a path 𝑣 1 𝑢 1 𝑢 2 𝑣 2 such that deg𝐺 (𝑢 1 ) = deg𝐺 (𝑢 2 ) = 2, then we call the graph 𝐺 (𝑣 1 , 𝑣 2 ) obtained from 𝐺 by contracting 𝑣 1 and 𝑣 2 and deleting {𝑢 1 , 𝑢 2 } a Type-1 𝐺-reducible graph. We note that if 𝐺 has order 𝑛, then the resulting graph 𝐺 (𝑣 1 , 𝑣 2 ) has order 𝑛 − 3.

Section 10.2. Domination and Forbidden Cycles

293

Definition 10.4 If a graph 𝐺 has a path 𝑥 1 𝑤 1 𝑤 2 𝑤 3 𝑥2 such that deg𝐺 (𝑤 2 ) = 2 and N𝐺 (𝑤 1 ) = N𝐺 (𝑤 3 ) = {𝑥1 , 𝑥2 , 𝑤 2 }, then we call the graph 𝐺 (𝑥1 , 𝑥2 ) obtained from 𝐺 by deleting {𝑤 1 , 𝑤 2 , 𝑤 3 } and adding the edge 𝑥 1 𝑥2 if the edge is not already present in 𝐺 a Type-2 𝐺-reducible graph. If 𝐺 has order 𝑛, then the resulting graph 𝐺 (𝑥 1 , 𝑥2 ) has order 𝑛 − 3. Definition 10.5 Let F4 = {𝐶4 }. For every 𝑖 ≡ 1 (mod 3) with 𝑖 > 4, we define the family F𝑖 as follows. A graph 𝐺 belongs to F𝑖 if and only if 𝛿(𝐺) ≥ 2 and there is a Type-1 or Type-2 𝐺-reducible graph in F𝑖−3 . By construction, for every 𝑖 ≥ 4 with 𝑖 ≡ 1 (mod 3), if 𝐺 ∈ F𝑖 , then 𝐺 has order 𝑖. Recall that in Chapter 6, we defined the family Bdom = {𝐵1 , 𝐵2 , . . . , 𝐵7 } of seven exceptional connected graphs with the property that if 𝐺 ∈ Bdom has order 𝑛, then 𝛿(𝐺) ≥ 2 and 𝛾(𝐺) > 52 𝑛. The family Bdom is featured in the statement of the classic McCuaig-Shepherd result, namely Theorem 6.18. We repeat here a drawing of the graphs in the family Bdom in Figure 10.2. The 4-cycle, labeled 𝐵1 in Figure 10.2, is a Type-1 𝐵𝑖 -reducible graph for all 𝑖 ∈ {2, 3, . . . , 7} and so 𝐵𝑖 ∈ F7 for all 𝑖 ∈ {2, 3, . . . , 7}. Note that the 4-cycle 𝐵1 is also a Type-2 𝐵𝑖 -reducible graph for 𝑖 ∈ {6, 7}. Henning et al. [477] observed that the family F7 consists of precisely the six graphs {𝐵2 , 𝐵3 , . . . , 𝐵7 } in the family Bdom of order 7. Hence, F4 = {𝐶4 }

and

F7 = Bdom \ {𝐶4 },

and |F4 | = 1 and |F7 | = 6. Equivalently, Bdom = F4 ∪ F7 .

𝐵1

𝐵4

𝐵2

𝐵5

𝐵3

𝐵6

𝐵7

Figure 10.2 The family Bdom

Definition 10.6 Let F≤13 = F4 ∪ F7 ∪ F10 ∪ F13 . As shown in [477], if 𝐺 ∈ F≤13 has order 𝑛, then 𝑛 ∈ {4, 7, 10, 13} and 𝛾(𝐺) = 13 (𝑛 + 2). A computer search shows that there are 28,076 non-isomorphic graphs in the family F≤13 . Furthermore, 41 of these graphs have bad-4-cut-vertices.

294 Therefore, let

Chapter 10. Domination and Forbidden Subgraphs

 Fforbid = 𝐺 ∈ F≤13 : bc4 (𝐺) = 0}

consist of the 28,035 graphs in the family F≤13 that do not have a bad-4-cut-vertex. We are now in a position to state the main result in [477]. Theorem 10.7 ([477]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then either 𝐺 ∈ Fforbid or 𝛾(𝐺) ≤ 83 𝑛 + 18 sc(𝐺) + 18 bc4 (𝐺). As a consequence of Theorem 10.7, we have the following result. Corollary 10.8 ([477]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, no special cycle, and no bad-4-cut-vertex, then 𝐺 ∈ Fforbid or 𝛾(𝐺) ≤ 38 𝑛. If 𝐺 is a graph with 𝛿(𝐺) ≥ 3, then 𝐺 contains no special cycle and no bad4-cut-vertex and 𝐺 ∉ Fforbid . Hence, Theorem 6.20 in Chapter 6 due to Reed is an immediate consequence of Corollary 10.8. As observed earlier, every graph in the family Fforbid has order at most 13, implying the following corollary of Theorem 10.7. Corollary 10.9 ([477]) If 𝐺 is a connected graph of order 𝑛 ≥ 14 with 𝛿(𝐺) ≥ 2, no special cycle, and no bad-4-cut-vertex, then 𝛾(𝐺) ≤ 38 𝑛. The McCuaig-Shepherd result given in Theorem 6.18 in Chapter 6 is an immediate corollary of Theorem 10.7. Corollary 10.10 ([477]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then 𝛾(𝐺) ≤ 25 𝑛, unless 𝐺 is one of the seven exceptional graphs in the family Bdom . Proof By Theorem 10.7, 𝐺 ∈ Fforbid or 𝛾(𝐺) ≤ 38 𝑛 + 18 sc(𝐺) + 18 bc4 (𝐺). Suppose that 𝐺 ∈ Fforbid . Recall that F≤13 = F4 ∪ F7 ∪ F10 ∪ F13 , where F4 = {𝐶4 } and Bdom = F4 ∪ F7 . If 𝐺 ∈ F10 , then 𝑛 = 10 and 𝛾(𝐺) = 13 (𝑛 + 2) = 4 = 25 𝑛, while if 𝐺 ∈ F13 , then 𝑛 = 13 and 𝛾(𝐺) = 13 (𝑛 + 2) = 5 < 25 𝑛. Hence, if 𝐺 ∈ Fforbid , then either 𝐺 ∈ Bdom or 𝛾(𝐺) ≤ 25 𝑛. Suppose next that 𝛾(𝐺) ≤ 38 𝑛 + 18 sc(𝐺) + 18 bc4 (𝐺). Since sc(𝐺) + bc4 (𝐺) ≤ 15 𝑛, this simplifies to 𝛾(𝐺) ≤ 38 𝑛 + 18 · 15 𝑛 = 25 𝑛. If 𝐺 is (𝐶4 , 𝐶5 )-free, then bc4 (𝐺) = sc(𝐺) = 0. Thus, we have the following consequence of Theorem 10.7. Corollary 10.11 ([477]) If 𝐺 is a (𝐶4 , 𝐶5 )-free connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then the following hold: (a) If 𝐺 ∉ Fforbid , then 𝛾(𝐺) ≤ 38 𝑛. (b) If 𝑛 ≥ 14, then 𝛾(𝐺) ≤ 38 𝑛. An (infinite) family of (𝐶4 , 𝐶5 )-free connected graphs achieving equality in the bound of Corollary 10.11(b) can be constructed as follows. Define an 8-key to be a graph of order 8 obtained from a 7-cycle by adding a pendant edge to one of its vertices. Define a unit to be a graph that is isomorphic to an 8-cycle or to an 8-key.

Section 10.2. Domination and Forbidden Cycles

295

We call a unit a cycle unit or a key unit if it is an 8-cycle or an 8-key, respectively. In each cycle unit, we select an arbitrary vertex 𝑣 and the two vertices at distance 3 from 𝑣 in the unit and we call these three vertices the link vertices of the cycle-unit, while in a key-unit we call the vertex of degree 1 the link vertex of the unit. Let Hdom be the family of all graphs 𝐺 such that either 𝐺 = 𝐶8 or 𝐺 can be obtained from the disjoint union of at least two units, each of which is a cycle unit or a key unit, by adding edges between link vertices so that the resulting graph is a (𝐶4 , 𝐶5 )-free connected graph where each added edge is a bridge of 𝐺. Every dominating set in 𝐺 contains at least three vertices from each unit of 𝐺. Thus, if 𝐺 has 𝑘 ≥ 2 units, then 𝑛 = 8𝑘 ≥ 16 and 𝛾(𝐺) ≥ 3𝑘 = 83 𝑛. By Corollary 10.11(b), 𝛾(𝐺) ≤ 38 𝑛. Consequently, 𝛾(𝐺) = 38 𝑛. A graph 𝐺 in the family Hdom with two cycle units and two key units is shown in Figure 10.3, where the highlighted vertices are the link vertices.

Figure 10.3 A graph 𝐺 in the family Hdom We note that if 𝐺 is 2-connected graph of order 𝑛 ≥ 5, then bc4 (𝐺) = 0 since 𝐺 can have no cut-vertices. Also, if deg𝐺 (𝑢) + deg𝐺 (𝑣) ≥ 5 for every two adjacent vertices 𝑢 and 𝑣 in a graph 𝐺, then sc(𝐺) = 0. Hence, we have the following corollary of Theorem 10.7. Corollary 10.12 ([477]) If 𝐺 is a 2-connected graph of order 𝑛 ≥ 14 and deg𝐺 (𝑢)+ deg𝐺 (𝑣) ≥ 5 for every two adjacent vertices 𝑢 and 𝑣 of 𝐺, then 𝛾(𝐺) ≤ 38 𝑛. Corollary 10.12 can be restated as follows. Corollary 10.13 ([477]) If 𝐺 is a 2-connected graph of order 𝑛 ≥ 14 and the set of vertices of degree 2 in 𝐺 is an independent set, then 𝛾(𝐺) ≤ 38 𝑛. That the bound of Corollary 10.12 is tight may be seen as follows. Let 𝑘 ≥ 2 be an integer and let G2conn be the family of all graphs that can be obtained from a 2-connected graph 𝐹 of order 2𝑘 that contains a perfect matching 𝑀 using the following procedure. Replace each edge 𝑢𝑣 in the matching 𝑀 by an 8-cycle 𝐶𝑢𝑣 : 𝑣 1 𝑣 2 . . . 𝑣 8 𝑣 1 with two chords 𝑣 1 𝑣 4 and 𝑣 2 𝑣 5 , where 𝑢 = 𝑣 8 and 𝑣 = 𝑣 6 . Let 𝐺 𝑢𝑣 denote the resulting subgraph and let 𝐺 denote the resulting 2-connected graph of order 𝑛 = 8𝑘. Every dominating set in 𝐺 contains at least three vertices from 𝐺 𝑢𝑣 for each edge 𝑢𝑣 ∈ 𝑀, and so 𝛾(𝐺) ≥ 3𝑘 = 38 𝑛. Since 𝐺 is a 2-connected graph of order 𝑛 ≥ 14 and deg𝐺 (𝑢) + deg𝐺 (𝑣) ≥ 5 for every two adjacent vertices 𝑢 and 𝑣 of 𝐺,

Chapter 10. Domination and Forbidden Subgraphs

296

by Corollary 10.12, we have 𝛾(𝐺) ≤ 83 𝑛. Consequently, 𝛾(𝐺) = 38 𝑛. A graph in the family G2conn with 𝑘 = 4 that is obtained from an 8-cycle 𝐹 is shown in Figure 10.4.

𝑣7 𝑣

𝑢

𝑣5

𝑣1 𝑣2

𝑣4 𝑣3

Figure 10.4 A graph in the family G2conn

Domination and Large Girth In this section, we study bounds on the domination number when there are no small cycles. The girth of a graph that contains a cycle is the length of a shortest cycle in the graph. In 1990 Brigham and Dutton [119] observed that the deletion of a shortest cycle 𝐶 of length 𝑔 from a graph 𝐺 of order 𝑛 with minimum degree at least 2 and girth at least 5 produces an isolate-free graph 𝐺 ′ of order 𝑛− 𝑔,implying  𝑔  by  𝑛 Ore’s 𝑔 bound given in Theorem 6.2 that 𝛾(𝐺) ≤ 𝛾(𝐺 ′ ) + 𝛾(𝐶) ≤ 𝑛−𝑔 + ≤ 2 3 2 − 6 . Theorem 10.14 ([119]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 5, then   𝑛 𝑔 𝛾(𝐺) ≤ − . 2 6 In 2005 Volkmann [735, 736] strengthened the bound of Theorem 10.14 slightly as follows. Theorem 10.15 ([735, 736]) For 𝑖 ∈ [2], if 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 5, then   𝑛 𝑔 𝑖+1 𝛾(𝐺) ≤ − − , 2 6 2 unless 𝐺 is a cycle or 𝐺 belongs to a set G𝑖 of graphs, where |G1 | = 2 and |G2 | = 40. Motivated by these results, in 2008 Rautenbach [650] proved the following result. Theorem 10.16 ([650]) For every 𝑘 ∈ N, there exists a finite set G𝑘 of graphs such that if 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 5, then 𝛾(𝐺) ≤

𝑛 𝑔 − − 𝑘, 2 6

unless 𝐺 is a cycle or 𝐺 belongs to the set G𝑘 . As discussed in Chapter 6, in 1996 Reed  [655] conjectured that if 𝐺 is a connected cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 13 𝑛 . Although the conjecture is false, all known

Section 10.2. Domination and Forbidden Cycles

297

counterexamples, including the constructions given by Kostochka and Stodolsky [538] and Kelmans [521], contain small cycles. Much discussion therefore centered on when Reed’s conjecture becomes true with the additional condition that the girth 𝑔 of the graph is sufficiently large. The first such result was presented in 2006 by Kawarabayashi et al. [518]. Recall that a 2-factor of a graph 𝐺 is a spanning 2-regular subgraph of 𝐺, that is, a collection of vertex-disjoint cycles that contain all the vertices of 𝐺. Theorem 10.17 ([518]) If 𝐺 is a connected cubic graph of order 𝑛 with girth 𝑔 ≥ 3 and a 2-factor, then   1 1 𝛾(𝐺) ≤ 𝑛. + 3 9⌊𝑔/3⌋ + 3 In 2009 Kostochka and Stodolsky [539] improved the bound of Theorem 10.17 for graphs without short cycles. Their proof exploits the ideas and techniques of vertex-disjoint path covers used in Reed’s seminal paper [655] and in addition uses intricate discharging arguments. Theorem 10.18 ([539]) If 𝐺 is a connected cubic graph of order 𝑛 with girth 𝑔 ≥ 3, then   1 8 𝛾(𝐺) ≤ + 𝑛. 3 3𝑔 2 In 2008 Rautenbach and Reed [651] and in 2012 Král et al. [540] established further upper bounds on the domination number of a cubic graph in terms of its order and girth. In 2008 Löwenstein and Rautenbach [572] proved a best possible upper bound on the domination number of graphs of minimum degree at least 2 and girth at least 5. Theorem 10.19 ([572]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 5, then ! 1 2 𝛾(𝐺) ≤ +  
𝑛. 3 3 3 𝑔+1 + 1 3

The bound of Theorem 10.19 is best possible for the union of cycles 𝐶3⌊ (𝑔+1)/3⌋+1 .   Since 3 𝑔+1 + 1 ≥ 𝑔 for 𝑔 ≥ 3, as an immediate consequence of Theorem 10.19, 3 we have the following result. The bound of Corollary 10.20 is best possible for the union of cycles 𝐶𝑔 , where 𝑔 ≡ 1 (mod 3). Corollary 10.20 ([572]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 5, then   1 2 𝛾(𝐺) ≤ + 𝑛. 3 3𝑔 As a corollary of Theorem 10.19, Löwenstein and Rautenbach [572] proved the following upper bound on the domination number of a cubic graph of girth at least 5.

Chapter 10. Domination and Forbidden Subgraphs

298

Theorem 10.21 ([572]) If 𝐺 is a cubic graph of order 𝑛 with girth 𝑔 ≥ 5, then   44 82 𝛾(𝐺) ≤ + 𝑛. 135 135𝑔 44 Since 135 + Theorem 10.21.

82 135𝑔


13 𝑛, as shown in Figure 10.5, where the highlighted vertices form a 𝛾-set of 𝐺.

Figure 10.5 The generalized Petersen graph 𝑃(7, 2) Another closely related 13 -conjecture for domination in cubic graphs can be attributed to Kostochka [536], who announced the following question in the open problem session at the Third International Conference on Combinatorics, Graph Theory and Applications, held at Elgersburg, Germany, March 2009: Is it true that the domination number of a bipartite cubic graph is at most one-third its order? Kostochka and Stodolsky commented in their paper in [539] that it would be interesting to answer this question. This intriguing question of Kostochka was posed seven years later as a formal conjecture by Henning in [459]. Conjecture 10.24 ([459]) If 𝐺 is a bipartite cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 1 3 𝑛.

Section 10.2. Domination and Forbidden Cycles

299

Both Conjectures 10.23 and 10.24 seem to be very challenging. The following conjecture, posed independently in 2006 by Kelmans [521] and in 2005 by Kostochka and Stodolsky [538], claims that Reed’s conjecture is true for 3-connected cubic graphs. Conjecture   10.25 ([521, 538]) If 𝐺 is a 3-connected cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 𝑛3 . At the same Elgersburg’s conference, Kostochka also posed the following question: Is it true that the vertex set of every bipartite cubic graph can be partitioned into three dominating sets? If this is true, then it would be a stronger result than the truth of Conjecture 10.24. This question of Kostochka was subsequently posed seven years later as a formal conjecture by Henning [459] in 2016. Conjecture 10.26 ([459]) The vertex set of every bipartite cubic graph can be partitioned into three dominating sets.

10.2.2

Total Domination Number

In this section, we present improved bounds on the total domination number of a graph when certain cycles are forbidden. Forbidden Induced 6-cycles and Minimum Degree Two Recall that Gtdom consists of all graphs 𝐺 that can be constructed from a connected graph 𝐹 of order at least 2 as follows: for each vertex 𝑣 of 𝐹, add a 6-cycle 𝐶𝑣 and join 𝑣 either to one vertex of 𝐶𝑣 or to two vertices at distance 2 apart on the cycle 𝐶𝑣 . The graph 𝐹 is called the underlying graph of 𝐺. A graph 𝐺 ∈ Gtdom with the underlying graph 𝐹 = 𝐶5 is illustrated in Figure 10.6. 𝐹 = 𝐶5

Figure 10.6 A graph in the family Gtdom As shown in Theorem 6.47, if 𝐺 is a connected graph of order 𝑛 > 14 and 𝛿(𝐺) ≥ 2, then the extremal graphs 𝐺 achieving the upper bound of 47 𝑛 for the total domination number are precisely the graphs in the family Gtdom . We note that every graph in the family Gdom contains an induced 6-cycle. Hence, it is a natural question to ask if the 47 -bound on the total domination can be improved if we forbid induced 6-cycles, or if this bound is still asymptotically best possible. In 2009 Henning and Yeo [485] showed that the 47 -bound can indeed be improved if

300

Chapter 10. Domination and Forbidden Subgraphs

we forbid induced 6-cycles. They proved a much stronger result. Some additional terminology is needed to present it. Definition 10.27 For vertex-disjoint subsets 𝑋 and 𝑌 of a graph 𝐺, an (𝑋, 𝑌 )-total dominating set, abbreviated (𝑋, 𝑌 )-TD-set, of 𝐺 is defined in [485] as a set 𝑆 of vertices of 𝐺 such that 𝑋 ∪ 𝑌 ⊆ 𝑆 and 𝑉 \ 𝑌 ⊆ N(𝑆). Thus, an (𝑋, 𝑌 )-TD-set is a set 𝑆 ⊆ 𝑉 such that 𝑆 contains 𝑋 ∪ 𝑌 and 𝑆 totally dominates the set 𝑉 \ 𝑌 . In particular, every vertex in 𝑋 is required to belong to 𝑆 and to have a neighbor in the (𝑋, 𝑌 )-TD-set 𝑆, while the vertices in 𝑌 are only required to belong to 𝑆 but not necessarily have a neighbor in 𝑆. The (𝑋, 𝑌 )-total domination number of 𝐺, denoted by 𝛾t (𝐺; 𝑋, 𝑌 ), is the minimum cardinality of an (𝑋, 𝑌 )-TD-set in 𝐺. Note that (∅, ∅)-TD-sets in 𝐺 are precisely the TD-sets in 𝐺. Hence, 𝛾t (𝐺) = 𝛾t (𝐺; ∅, ∅). As remarked in [485], the concept of an (𝑋, 𝑌 )-TD-set is related to the concept of restricted domination in graphs when certain vertices are specified to belong to the dominating set, introduced in 1997 by Sanchis in [664] and studied further, for example, in [351, 455]. Restricted total domination in graphs was first introduced and studied by Henning in [457]. Definition 10.28 Let 𝐺 be a graph and let 𝑋 and 𝑌 be disjoint vertex sets in 𝐺. A vertex 𝑥 in 𝐺 is called an (𝑋, 𝑌 )-cut-vertex in [485] if the following hold: (a) The graph 𝐺 − 𝑥 contains a component which is an induced 6-cycle 𝐶 𝑥 and which does not contain any vertices from 𝑋 or 𝑌 . (b) The vertex 𝑥 is adjacent to exactly one vertex on 𝐶 𝑥 or to exactly two vertices at distance 2 apart on 𝐶 𝑥 . Next we define a bad-6-cut-vertex in a graph. Definition 10.29 When 𝑋 = 𝑌 = ∅, we call an (𝑋, 𝑌 )-cut-vertex of 𝐺 a bad-6-cutvertex of 𝐺 and we denote the number of bad-6-cut-vertices in 𝐺 by bc(𝐺; 𝑋, 𝑌 ) (standing for “bad cut-vertex”) or simply by bc6 (𝐺). Thus, bc6 (𝐺) is the number of bad-6-cut-vertices in 𝐺. In the graph 𝐺 shown in Figure 10.6, for example, when the underlying graph 𝐹 of 𝐺 is a 5-cycle, let 𝑋 and 𝑌 be disjoint vertex sets in 𝐺 such that 𝑋 ∪ 𝑌 ⊆ 𝑉 (𝐹). Then, each vertex in 𝑉 (𝐹) is an (𝑋, 𝑌 )-cut-vertex of 𝐺 and bc(𝐺; 𝑋, 𝑌 ) = 5. In particular, when 𝑋 = 𝑌 = ∅, we have bc6 (𝐺) = 5. Next we define what is meant by a 𝐺-reducible graph, and thereafter define a family of Ftdom,𝑖 of forbidden graphs. Definition 10.30 If there is a path 𝑣 1 𝑢 1 𝑢 2 𝑢 3 𝑣 3 in a graph 𝐺 such that deg(𝑢 1 ) = deg(𝑢 2 ) = deg(𝑢 3 ) = 2 in 𝐺, then we call the graph obtained from 𝐺 by contracting 𝑣 1 and 𝑣 3 and deleting {𝑢 1 , 𝑢 2 , 𝑢 3 } a 𝑃5 -reduction of 𝐺. Note that it is possible that 𝑣 1 𝑣 3 is an edge of 𝐺. Definition 10.31 For 𝑘 ∈ {3, 5, 6}, let Ftdom,𝑘 = {𝐶 𝑘 }. For notational convenience, let Ftdom,4 = ∅. For every 𝑖 > 6, we define Ftdom,𝑖 as follows. A graph 𝐺 belongs to Ftdom,𝑖 if and only if 𝛿(𝐺) ≥ 2 and 𝐺 has a 𝑃5 -reduction to a graph that belongs to Ftdom,𝑖−4 .

Section 10.2. Domination and Forbidden Cycles

(a)

(b)

(c)

(d)

301

(e)

(f)

Figure 10.7 The family Ftdom,9

To illustrate Definition 10.31, the six graphs in the family Ftdom,9 are shown in Figure 10.7. ′ and 𝐶 ′′ , shown in Figure 6.19 in Chapter 6, belong to the The graphs 𝐶10 10 family Ftdom,10 . Every graph in the family Ftdom,𝑖 is a connected graph of order 𝑖. Further, the total domination number of a graph in the family Ftdom,𝑖 cannot be too large, as the following lemma shows.   Lemma 10.32 ([485]) For every integer 𝑖 ≥ 3, if 𝐺 ∈ Ftdom,𝑖 , then 𝛾t (𝐺) ≤ 𝑖+2 2 . Let 𝐺 7 be the graph shown in Figure 10.8 (and also shown in Figure 6.17(a)).

Figure 10.8 The graph 𝐺 7 We now define a family Ftdom of (forbidden) graphs. Let I = {3, 5, 6, 7, 9, 10, 14, 18} and let ! Ø Ftdom = {𝐾2 , 𝐺 7 } ∪ Ftdom,𝑖 . 𝑖∈I

 ′ , 𝐶 ′′ . The Recall that in Section 6.3.2, we defined Btdom = 𝐶3 , 𝐶5 , 𝐶6 , 𝐶10 , 𝐶10 10 following result is proven in [485]. Lemma 10.33 ([485]) If 𝐺 ∈ Ftdom has order 𝑛, then the following hold:   (a) 𝛾t (𝐺) ≤ 12 𝑛 + 1. (b) If 𝐺 ≠ 𝐾2 and 𝐺 ∉ Btdom , then 𝛾t (𝐺) ≤ 47 𝑛. As before, let 𝑋 and 𝑌 be vertex-disjoint sets in a graph 𝐺. Let 𝛿1 (𝐺; 𝑋, 𝑌 ) denote the number of vertices of degree 1 in 𝐺 that do not belong to the set 𝑌 , and let 𝛿2,1 (𝐺; 𝑋, 𝑌 ) denote the number of vertices of degree 2 in 𝐺 that do not belong to 𝑋 ∪ 𝑌 and are adjacent to a degree-1 vertex in 𝐺 that does not belong to 𝑋 ∪ 𝑌 . We are now in a position to state the main result in [485]. We remark that the proof of this result given by Henning and Yeo [485] relies heavily on an interplay between total domination in graphs and transversals in hypergraphs.

Chapter 10. Domination and Forbidden Subgraphs

302

Theorem 10.34 ([485]) If 𝐺 is a connected graph of order 𝑛 ≥ 2 and 𝑋 and 𝑌 are two vertex-disjoint sets of 𝐺, then either 𝑋 = 𝑌 = ∅ and 𝐺 ∈ Ftdom or 11𝛾t (𝐺; 𝑋, 𝑌 ) ≤ 6𝑛 + 8|𝑋 | + 5|𝑌 | + 2bc(𝐺; 𝑋, 𝑌 ) + 2𝛿1 (𝐺; 𝑋, 𝑌 ) + 2𝛿2,1 (𝐺; 𝑋, 𝑌 ). If 𝐺 is a graph with 𝛿(𝐺) ≥ 2 and 𝑋 and 𝑌 are vertex-disjoint sets in 𝐺, then 𝛿1 (𝐺; 𝑋, 𝑌 ) = 𝛿2,1 (𝐺; 𝑋, 𝑌 ) = 0. Hence, setting 𝑋 = 𝑌 = ∅ in Theorem 10.34, we have the following immediate consequence of Theorem 10.34. Theorem 10.35 ([485]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then either 𝐺 ∈ Ftdom or 6 2 𝛾t (𝐺) ≤ 11 𝑛 + 11 bc6 (𝐺). As a consequence of Lemma 10.33 and Theorem 10.35, we have the following result, which is a restatement of Theorem 6.46. Corollary 10.36 ([485]) If 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then 𝛾t (𝐺) ≤ 47 𝑛, unless 𝐺 is one of the six exceptional graphs in the family Btdom . Proof Let 𝐺 be a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2. We note that the number of bad-6-cut-vertices in a graph 𝐺 of order 𝑛 is bc6 (𝐺) ≤ 17 𝑛. By Theorem 10.35, either 𝐺 ∈ Ftdom or 𝛾t (𝐺) ≤

6 11 𝑛

+

2 11 bc6 (𝐺)

≤

6 11 𝑛

+

2 11

· 17 𝑛 = 47 𝑛.

If 𝐺 ∈ Ftdom and 𝐺 ∉ Btdom , then by Lemma 10.33(b), we have 𝛾t (𝐺) ≤ Hence, 𝛾t (𝐺) ≤ 47 𝑛, unless 𝐺 ∈ Btdom .

4 7 𝑛.

If 𝐺 contains no induced 6-cycle, then bc6 (𝐺) = 0. Since every graph in the family Ftdom has order at most 18, as a consequence of Theorem 10.35, we have the following result. Theorem 10.37 ([485]) If 𝐺 is a connected 𝐶6 -free graph of order 𝑛 ≥ 19 with 6 𝛿(𝐺) ≥ 2, then 𝛾t (𝐺) ≤ 11 𝑛. Thus, by Theorem 10.37, if we forbid induced 6-cycles, then the upper bound on the total domination number of a graph with minimum degree at least 2 can be 6 improved from the 47 -bound in Theorem 6.46 to a 11 -bound. To illustrate the tightness of Theorem 10.37, let Gtdom,6 be the family of graphs 𝐺 that can be constructed from a connected 𝐶6 -free graph 𝐹 of order at least 2 as follows: for each vertex 𝑣 of 𝐹, add a 10-cycle 𝐶𝑣 and join 𝑣 to one vertex of 𝐶𝑣 . The graph 𝐹 is called the underlying graph of 𝐺. Each graph 𝐺 ∈ Gtdom,6 is a connected 6 𝐶6 -free graph of order 𝑛 with 𝛾t (𝐺) = 11 𝑛. A graph 𝐺 ∈ Gtdom,6 when the underlying graph 𝐹 = 𝐶4 is illustrated in Figure 10.9. Forbidden 4- and 6-cycles and Minimum Degree Three In the previous section, we showed that if we forbid induced 6-cycles, then the upper bound on the total domination number of a graph with minimum degree 2 can be

Section 10.2. Domination and Forbidden Cycles

303

𝐹 = 𝐶4

Figure 10.9 A graph in the family Gtdom,6

6 improved from a 47 -bound (see Theorem 6.46) to a 11 -bound (see Theorem 10.37). In this section, we study the effect of forbidden 4- and 6-cycles on the total domination number of a graph with minimum degree at least 3. As shown in Theorems 6.52 and 6.59, if 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 12 𝑛, with equality if and only if 𝐺 ∈ Gcubic ∪ Hcubic or 𝐺 is the generalized Petersen graph 𝑃(8, 3), where the families Gcubic and Hcubic are defined in Section 6.3.3. An example of a graph 𝐺 4 ∈ Gcubic and a graph 𝐻4 ∈ Hcubic is illustrated in Figure 10.10(a) and (b), respectively, while the generalized Petersen graph 𝑃(8, 3) is shown in Figure 10.10(c). We note that every vertex that belongs to a graph in the family Gcubic ∪ Hcubic belongs to an induced 6-cycle, as does every vertex of the generalized Petersen graph 𝑃(8, 3). We also note that every vertex that belongs to a graph in the family Gcubic ∪ Hcubic belongs to a 4-cycle. Hence, it is a natural question to ask if the 12 -bound on the domination number can be significantly improved if we forbid induced 4-cycles and/or forbid induced 6-cycles, or if this 12 -bound is still asymptotically best possible.

(a) 𝐺 4

(b) 𝐻4

(c) 𝑃(8, 3)

Figure 10.10 The graphs 𝐺 4 ∈ Gcubic , 𝐻4 ∈ Hcubic , and 𝑃(8, 3)

In 2021 Henning and Yeo [494] made partial progress on this problem by relaxing the condition, simply forbidding all 4-cycles and all 6-cycles (not necessarily induced), to show that that the 12 -bound can indeed be improved. To present their result, we shall need some additional hypergraph terminology. A linear hypergraph is one in which every two distinct edges intersect in at most one vertex. Two edges in a hypergraph 𝐻

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Chapter 10. Domination and Forbidden Subgraphs

overlap if they intersect in at least two vertices. A linear hypergraph therefore has no overlapping edges. A finite affine plane AG(2, 𝑞) of dimension 2 and order 𝑞 ≥ 2 is a collection of 𝑞 2 points and 𝑞 2 + 𝑞 lines such that each line contains 𝑞 points and each point is contained in 𝑞 + 1 lines. The affine plane AG(2, 3) of dimension 2 and order 3 is illustrated in Figure 10.11. This is equivalent to a linear 3-uniform 4-regular hypergraph 𝐻 on nine vertices, where the lines of AG(2, 3) correspond to the 3-edges of 𝐻.

Figure 10.11 The affine plane AG(2, 3) The authors in [494] showed that if 𝐻 is a linear hypergraph that does not contain a subhypergraph isomorphic to the affine plane AG(2, 3) of order 3 with two vertices deleted, then 17𝜏(𝐻) ≤ 5𝑛(𝐻) + 3𝑚(𝐻). As a consequence of this hypergraph result, they proved the following graph theory result using the interplay between total domination in graphs and transversals in hypergraphs, where 𝐺 12 is the graph shown in Figure 10.12.

Figure 10.12 The graph 𝐺 12

Theorem 10.38 ([494]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, no 4-cycle, and 8 no 𝐺 12 subgraph, then 𝛾t (𝐺) ≤ 17 𝑛. We remark that in the statement of Theorem 10.38, all 4-cycles are forbidden, even if they are not induced. Further, both conditions, namely the conditions that the graph contains no 4-cycle and no 𝐺 12 as a subgraph, are essential. As observed earlier, there are infinitely many connected cubic graphs 𝐺 of order 𝑛 that contain 4-cycles and satisfy 𝛾t (𝐺) = 12 𝑛. Moreover, the generalized Petersen graph 𝐺 = 𝑃(8, 3) contains no 4-cycle but does contain 𝐺 12 as a subgraph and satisfies 𝛾t (𝐺) = 12 𝑛. We note that if 𝐺 is a graph that contains no 4-cycles and no 6-cycles, then 𝐺 12 is not a subgraph

Section 10.2. Domination and Forbidden Cycles

305

of 𝐺. Hence, as an immediate consequence of Theorem 10.38, we have the following result. Theorem 10.39 ([494]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, no 4-cycle, and 8 no 6-cycle, then 𝛾t (𝐺) ≤ 17 𝑛. We close this section on forbidden 4- and 6-cycles with the following two conjectures. Conjecture 10.40 ([496]) If 𝐺 is not the generalized Petersen graph 𝑃(8, 3) and 𝐺 8 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 3 and no 4-cycle, then 𝛾t (𝐺) ≤ 17 𝑛. If Conjecture 10.40 is true, then the upper bound is tight as may be seen by considering the following family Gtdom,4 of graphs with minimum degree at least 3 and with no 4-cycles, that can be constructed as follows: let 𝐹 be a connected graph with 𝛿(𝐹) ≥ 2 that contains no 4-cycles and for each vertex 𝑣 of 𝐹, add a copy 𝐺 𝑣 of the generalized Petersen graph 𝑃(8, 3) and join 𝑣 to one vertex of 𝐺 𝑣 . A graph 𝐺 in the family Gtdom,4 is illustrated in Figure 10.13.

𝐹

Figure 10.13 A graph 𝐺 ∈ Gtdom,4

Proposition 10.41 ([490]) If 𝐺 ∈ Gtdom,4 has order 𝑛, then 𝐺 is a connected graph 8 with 𝛿(𝐺) ≥ 3, no 4-cycle, and 𝛾t (𝐺) = 17 𝑛. The following conjecture implies that if we forbid induced 6-cycles, then the upper bound on the total domination number can be improved from a 12 -bound (see Theorem 6.52) to a 49 -bound. Conjecture 10.42 If 𝐺 is a connected 𝐶6 -free graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 49 𝑛. If Conjecture 10.42 is true, then the bound is best possible. For example, the graph 𝐺 18 of order 𝑛 = 18, shown in Figure 10.14, has no induced 6-cycle and satisfies 𝛾t (𝐺) = 8 = 49 𝑛, where the red vertices are an example of a 𝛾t -set of 𝐺 18 .

Chapter 10. Domination and Forbidden Subgraphs

306

Figure 10.14 The graph 𝐺 18

Forbidden 4-cycles and Minimum Degree Four In the previous section, we showed that if we forbid 4- and 6-cycles, then the upper bound on the total domination number of a graph with minimum degree 3 can be 8 improved from a 12 -bound (see Theorem 6.52) to an 17 -bound (see Theorem 10.39). In this section, we study the effect of forbidden 4-cycles on the total domination number of a graph with minimum degree at least 4. As shown in Theorem 6.65, if 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 4, then 𝛾t (𝐺) ≤ 37 𝑛, with equality if and only if 𝐺 is the bipartite complement of the Heawood graph, which is shown in Figure 10.15(b) along with the Heawood graph in Figure 10.15(a). We note that the Heawood graph and its bipartite complement are also shown in Chapter 6.

(a) The Heawood graph

(b) The bipartite complement

Figure 10.15 The Heawood graph and its bipartite complement

Note that every vertex in the bipartite complement of the Heawood graph belongs to a 4-cycle. It is therefore a natural question to ask whether the upper bound of 37 𝑛 on the total domination number of a graph with minimum degree at least 4 can be improved if we restrict the graph to contain no 4-cycles, or if the 37 -bound is asymptotically best possible even with 4-cycles forbidden. In 2020 Henning and Yeo [492] proved the following upper bound on the transversal number of a linear 4-uniform hypergraph. Theorem 10.43 ([492]) If 𝐻 is a 4-uniform linear hypergraph of order 𝑛 and size 𝑚, then 𝜏(𝐻) ≤ 15 (𝑛 + 𝑚). As a consequence of Theorem 10.43, we have the following result which improves the 37 -bound on the total domination number of a graph of order 𝑛 with minimum degree at least 4 that contains no 4-cycle to a 25 -bound.

Section 10.2. Domination and Forbidden Cycles

307

Theorem 10.44 ([492]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 4 and no 4-cycle, then 𝛾t (𝐺) ≤ 52 𝑛. Proof Let 𝐺 be a graph of order 𝑛 with 𝛿(𝐺) ≥ 4 that contains no 4-cycles. Consider the open neighborhood hypergraph ONH(𝐺) of 𝐺. Note that each edge of ONH(𝐺) has size at least 4. Since 𝐺 contains no 4-cycle, the hypergraph ONH(𝐺) contains no overlapping edges and is therefore a linear hypergraph. Consider the hypergraph 𝐻 obtained from ONH(𝐺) by shrinking all edges of ONH(𝐺), if necessary, to edges of size 4 to produce a 4-uniform linear hypergraph 𝐻 of order 𝑛(𝐻) = 𝑛 and size 𝑚(𝐻) = 𝑛. Since every transversal of 𝐻 is a transversal of ONH(𝐺), by Observation 6.56 Theorem 10.43, we have 𝛾t (𝐺) = 𝜏(ONH(𝐺)) ≤ 𝜏(𝐻) ≤
and 1 1 + 𝑚(𝐻) = 𝑛(𝐻) (𝑛 + 𝑛) = 25 𝑛. 5 5 That the bound in Theorem 10.44 is best possible may be seen by the 4-regular bipartite graph 𝐺 30 of order 𝑛 = 30 illustrated in Figure 10.16, which has no 4-cycle and satisfies 𝛾t (𝐺 30 ) = 12 = 25 𝑛. We note that the graph 𝐺 30 is the incidence bipartite graph of the linear 4-uniform hypergraph obtained by removing an arbitrary vertex from the affine plane AG(2, 4) of order 4.

Figure 10.16 The graph 𝐺 30

Total Domination and Large Girth In this section, we study bounds on the total domination number when there are no small cycles. In 2009 Haynes and Henning [423] provided the following upper bound on the total domination number of a graph in terms of its girth 𝑔 and order 𝑛. Theorem 10.45 ([423]) If 𝐺 ≠ 𝐶𝑛 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 7, then 𝛾t (𝐺) ≤ 23 𝑛 − 16 𝑔 unless 𝐺 = 𝐻14 , where 𝐻14 is the graph shown in Figure 10.17, in which case 𝛾t (𝐺) = 8 = 23 𝑛 + 13 − 16 𝑔. If we allow small girth, then we have the following more general result in 2008 by Henning and Yeo [484]. The proof of Theorem 10.46 relies heavily on the interplay between total domination in graphs and transversals in hypergraphs.

Chapter 10. Domination and Forbidden Subgraphs

308

Figure 10.17 The graph 𝐻14

Theorem 10.46 ([484]) If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 3,
then 𝛾t (𝐺) ≤ 12 + 𝑔1 𝑛.       For 𝑛 ≥ 3, we observe that 𝛾t (𝐶𝑛 ) = 𝑛2 + 𝑛4 − 𝑛4 . Thus, if 𝑛 ≡ 2 (mod 4)
1 1 and 𝐺 = 𝐶𝑛 , then 𝐺 has order 𝑛, girth 𝑔 = 𝑛, and 𝛾t (𝐺) = 𝑛+2 2 = 2 + 𝑔 𝑛. Hence, the bound in Theorem 10.46 is tight for cycles of length congruent to 2 modulo 4. If 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 3, then by Theorem 10.46,

𝛾t (𝐺) ≤ 12 + 𝑔1 𝑛 = 23 𝑛 − 16 𝑛 − 6𝑛 𝑔 . Therefore, Theorem 10.46 improves on the bound of Theorem 10.45 when 𝑛 − 6𝑛/𝑔 > 𝑔, that is, when 𝑛 ≥ 𝑔 2 /(𝑔 − 6). The upper bounds in both Theorem 10.45 and Theorem 10.46 on the total domination number of a graph with minimum degree at least 2 in terms of its order and girth were subsequently improved in 2012 by Henning and Yeo [487]. To state their result, recall that in Section 10.2.2 we defined the family Ftdom,𝑖 for 𝑖 ≥ 3. For 𝑖 ≥ 1 an integer, the family Htdom,𝑖 of graphs is defined as follows: ! ! ! 𝑖 𝑖+1 2𝑖+2 Ø Ø Ø Htdom,𝑖 = Ftdom,4𝑘+3 ∪ Ftdom,4𝑘+1 ∪ Ftdom,4𝑘+2 ∪ {𝐾2 }, 𝑘=0

𝑘=1

and let Htdom =

𝑘=1 ∞ Ø

Htdom,𝑖 .

𝑖=1

Every graph in the family Htdom,𝑖 has order at most 8𝑖 + 10. Note that the graph 𝐾2 has order 𝑛 = 2 and 𝛾t (𝐺) = 2 = 𝑛+2 2 . As a consequence of Lemma 10.32, we have the following result.   Lemma 10.47 ([485]) If 𝐺 ∈ Htdom has order 𝑛, then 𝛾t (𝐺) ≤ 𝑛+2 2 . We are now in a position to state the following result from [487]. Theorem 10.48 ([487]) For 𝑖 ≥ 1 an integer, if 𝐺 is a connected graph of order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 4𝑖 + 3, then either 𝐺 ∈ Htdom,𝑖 or   2𝑖 + 4 𝛾t (𝐺) ≤ 𝑛. 4𝑖 + 7 As a consequence of Theorem 10.48, the upper bounds given in Theorem 10.45 and Theorem 10.46 can be improved as follows.

Section 10.2. Domination and Forbidden Cycles

309

Theorem 10.49 ([487]) If 𝐺 is a connected graph order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 3, then   𝑛 𝑛 𝛾t (𝐺) ≤ + max 1, . 2 2(𝑔 + 1) Proof Let 𝐺 be a connected graph order 𝑛 with 𝛿(𝐺) ≥ 2 and girth 𝑔 ≥ 3. When 𝑔 ≤ 6, we have     1 1 1 1 4 + 𝑛≥ + 𝑛 = 𝑛. 2 2(𝑔 + 1) 2 14 7 By Theorem 6.46, we know that 𝛾t (𝐺) ≤ 47 𝑛, unless 𝐺 is one of the six exceptional graphs in the family Btdom . However, each of the six graphs 𝐺 in the family Btdom satisfies 𝛾t (𝐺) ≤ 12 𝑛 + 1. Hence, if 𝑔 ≤ 6, then the result of the theorem is immediate. Therefore, we may assume that 𝑔 ≥ 7. Since 𝑔 ≥ 7, we can write 𝑔 = 4𝑖 + 𝑗 for some integers 𝑖 ≥ 1 and 𝑗 ∈ {3, 4, 5, 6}. In particular, 𝑔 ≥ 4𝑖 + 3. By Theorem 10.48, either 𝐺 ∈ Htdom , in which case 𝛾t (𝐺) ≤ 12 𝑛 + 1 by Lemma 10.47, or  𝛾t (𝐺) ≤

   2𝑖 + 4 1 1 𝑛≤ + 𝑛. 4𝑖 + 7 2 2(𝑔 + 1)

To illustrate the tightness of Theorems 10.48 and 10.49, let Hgirth be the family of all graphs that can be obtained from a connected graph 𝐻 of order at least 2 and girth at least 4𝑖 + 3 for 𝑖 ≥ 1 as follows: for each vertex 𝑣 of 𝐻, add a (4𝑖 + 6)-cycle and join 𝑣 to exactly one vertex of this cycle. Each graph 𝐺 ∈ Hgirth is a connected graph of order 𝑛 = (4𝑖 + 7)|𝑉 (𝐻)| and girth at least 4𝑖 + 3 with   2𝑖 + 4 𝛾t (𝐺) = (2𝑖 + 4)|𝑉 (𝐻)| = 𝑛. 4𝑖 + 7 A graph 𝐺 in the family Hgirth is illustrated in Figure 10.18. If 𝐻 has girth at least 𝑛 4𝑖 + 6, then the resulting graph 𝐺 has girth 𝑔 = 4𝑖 + 6 and satisfies 𝛾t (𝐺) = 𝑛2 + 2(𝑔+1) , thus achieving equality in the bound of Theorem 10.49.

𝐻

𝐶4𝑖+6

𝐶4𝑖+6

𝐶4𝑖+6

𝐶4𝑖+6

Figure 10.18 A graph in the family Hgirth
If 𝐺 is a graph of order 𝑛 and girth 𝑔, then 𝑛 ≥ 𝑔, implying that 12 + 𝑔1 𝑛 ≥ 12 𝑛 + 1. Therefore, Theorem 10.49 is a stronger result than Theorem 10.46.

Chapter 10. Domination and Forbidden Subgraphs

310

10.2.3

Independent Domination Number

In this section, we present improved bounds on the independent domination number of a cubic graph when certain cycles are forbidden. Forbidden 4-cycles The graphs 𝐾3,3 and 𝐶5 □ 𝐾2 are shown in Figure 10.19(a) and (b), respectively.

(a) 𝐾3,3

(b) 𝐶5 □ 𝐾2

Figure 10.19 The graphs 𝐾3,3 and 𝐶5 □ 𝐾2 In this section, we consider Conjecture 6.95 which we presented in Section 6.4.3. We repeat here the statement of the conjecture. Conjecture 10.50 ([236]) If 𝐺 ∉ {𝐾3,3 , 𝐶5 □ 𝐾2 } is a connected cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 38 𝑛. We show that Conjecture 10.50 is true if we forbid 4-cycles. In order to prove this result, Dorbec et al. [236] gave a stronger result. Recall that a subcubic graph is a graph with maximum degree at most 3. For a subcubic graph 𝐺, let 𝑛𝑖 (𝐺) denote the number of vertices of degree 𝑖 in 𝐺 for 𝑖 ∈ [3] 0 . Theorem 10.51 ([236]) If 𝐺 is a 𝐾2,3 -free subcubic graph with no (𝐶5 □ 𝐾2 )component, then 8𝑖(𝐺) ≤ 8𝑛0 (𝐺) + 5𝑛1 (𝐺) + 4𝑛2 (𝐺) + 3𝑛3 (𝐺). The proof of Theorem 10.51 given in [236] uses a method of weighting vertices along with detailed counting arguments, which we do not discuss here. The condition in Theorem 10.51 that 𝐺 does not contain an induced 𝐾2,3 cannot be dropped. For example, the graph 𝐺 obtained by attaching a pendant edge to a vertex of degree 2 in 𝐾2,3 (or equivalently by deleting two adjacent edges in 𝐾3,3 ) has 𝑖(𝐺) = 3, 𝑛3 (𝐺) = 3, 𝑛2 (𝐺) = 2, and 𝑛1 (𝐺) = 1, implying in this case that 8𝑖(𝐺) = 24 > 22 = 8 · 0 + 5 · 1 + 4 · 2 + 3 · 3 = 8𝑛0 (𝐺) + 5𝑛1 (𝐺) + 4𝑛2 (𝐺) + 3𝑛3 (𝐺). In the special case of Theorem 10.51 when 𝐺 is a connected cubic graph, we have the following result showing that Conjecture 10.50 is true if there is no subgraph isomorphic to 𝐾2,3 . Theorem 10.52 ([236]) If 𝐺 ≠ 𝐶5 □ 𝐾2 is a connected 𝐾2,3 -free cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 38 𝑛.

Section 10.2. Domination and Forbidden Cycles

311

1 defined in Section 6.4.3 are 𝐾2,3 -free, Graphs that belong to the family Fcubic and by Proposition 6.96, achieve this bound exactly. An example of a graph in the 1 family Fcubic is illustrated in Figure 10.20. Hence, the bound in Theorem 10.52 is achieved for infinitely many graphs.

1 Figure 10.20 A graph in the family Fcubic

As an immediate consequence of Theorem 10.52, we observe that Conjecture 10.50 is true if we forbid induced 4-cycles. Corollary 10.53 ([236]) If 𝐺 is a connected 𝐶4 -free cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 38 𝑛. As a consequence of Theorem 10.51, Dorbec et al. [236] proved the following upper bound on the domination number of a subcubic graph. Theorem 10.54 ([236]) If 𝐺 is a subcubic graph, then 8𝛾(𝐺) ≤ 8𝑛0 (𝐺) + 5𝑛1 (𝐺) + 4𝑛2 (𝐺) + 3𝑛3 (𝐺). A special case of Theorem 10.54 when 𝐺 is a cubic graph of order 𝑛 is Reed’s result [655] that 𝛾(𝐺) ≤ 38 𝑛 (see Corollary 10.55 below). This provides a different proof to that given by Reed’s vertex-disjoint path proof technique, where he chooses a vdp-cover of a graph to prove the 38 -bound on the domination number of a cubic graph. However, the new proof given in [236] for cubic graphs does not extend to general graphs with minimum degree at least 3 (where vertices of degree greater than 3 are thrown into the mix). Corollary 10.55 ([655]) If 𝐺 is a cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 38 𝑛. If 𝐺 is a subcubic graph of order 𝑛 and size 𝑚 with 𝑖 isolated vertices, then 8𝑛0 (𝐺) + 5𝑛1 (𝐺) + 4𝑛2 (𝐺) + 3𝑛3 (𝐺) = 6𝑛 − 2𝑚 + 2𝑖. Hence, as an immediate consequence of Theorem 10.54 we have the following 1999 result due to Fisher et al. [303] and Rautenbach [650]. Corollary 10.56 ([303, 650]) If 𝐺 is a subcubic graph of order 𝑛 and size 𝑚 with 𝑖 isolated vertices, then 4𝛾(𝐺) ≤ 3𝑛 − 𝑚 + 𝑖.

312

Chapter 10. Domination and Forbidden Subgraphs

Bipartite Graphs In this section, we consider upper bounds on the independent domination number of a bipartite cubic graph. As a special case of Theorem 6.90, we have the following upper bound on the independent domination number. Theorem 10.57 If 𝐺 is a connected bipartite cubic graph of order 𝑛, then 𝑖(𝐺) ≤ 1 2 𝑛, with equality if and only if 𝐺 = 𝐾3,3 . A natural question is whether we can improve the 12 -bound in Theorem 10.57 if we exclude the exceptional graph 𝐾3,3 . In 2013 Goddard and Henning [352] posed the following conjecture. Conjecture 10.58 ([352]) If 𝐺 ≠ 𝐾3,3 is a connected bipartite cubic graph of 4 order 𝑛, then 𝑖(𝐺) ≤ 11 𝑛. Southey [683], as part of his PhD thesis, confirmed by computer search that Conjecture 10.58 is true when 𝑛 ≤ 26. The upper bound in Conjecture 10.58 is achieved, 4 for example, by the bipartite cubic graph 𝐺 22 of order 𝑛 = 22 with 𝑖(𝐺 22 ) = 8 = 11 𝑛 shown in Figure 10.21, where the highlighted vertices form an 𝑖-set of 𝐺 22 .

Figure 10.21 The bipartite cubic graph 𝐺 22 In 2014 Henning et al. [468] showed that Conjecture 10.58 is true if the girth is at least 6. Theorem 10.59 ([468]) If 𝐺 is a bipartite cubic graph of order 𝑛 with girth at 4 least 6, then 𝑖(𝐺) ≤ 11 𝑛. In 2018 Abrishami and Henning [3] showed that the bipartite condition imposed on the cubic graph in the statement of Theorem 10.59 can be relaxed. Theorem 10.60 ([3]) If 𝐺 is a cubic graph of order 𝑛 with girth at least 6, then 4 𝑖(𝐺) ≤ 11 𝑛. Subsequently, in 2019 Abrishami and Henning [4] extended the result of Theo5 4 rem 10.60 and improved the 11 -bound to a 14 -bound. Theorem 10.61 ([4]) If 𝐺 is a cubic graph of order 𝑛 with girth at least 6, then 5 𝑖(𝐺) ≤ 14 𝑛.

Section 10.2. Domination and Forbidden Cycles

313

The condition in Theorem 10.61 that 𝐺 has girth at least 6 is essential. Indeed, if 𝐺 is the cubic bipartite graph 𝐺 22 of order 𝑛 = 22 shown in Figure 10.21, then 𝐺 4 5 has girth 4 and 𝑖(𝐺) = 8 = 11 𝑛 > 14 𝑛. The result of Theorem 10.59 was subsequently improved in 2019 by Brause and Henning [109]. In order to state their result, they defined three graphs 𝐵4 , 𝐵6 , and 𝐵12 called “bad graphs” shown in Figure 10.22. Further, they defined a bad component of a graph 𝐺 to be a component of 𝐺 isomorphic to 𝐵4 , 𝐵6 , or 𝐵12 .

(a) 𝐵4

(b) 𝐵6

(c) 𝐵12

Figure 10.22 The “bad graphs” 𝐵4 , 𝐵6 , and 𝐵12 Given a graph 𝐺, let 𝑏 1 (𝐺) be the number of components of 𝐺 isomorphic to 𝐵6 or 𝐵12 , and let 𝑏 2 (𝐺) be the number of components of 𝐺 isomorphic to 𝐵4 . Brause and Henning [109] established the following upper bound on the independent domination number of a bipartite subcubic graph that does not have an induced subgraph isomorphic to 𝐾2,3 , where as before we let 𝑛𝑖 (𝐺) denote the number of vertices of degree 𝑖 in 𝐺 for 𝑖 ∈ [3] 0 . Theorem 10.62 ([109]) If 𝐺 is a 𝐾2,3 -free bipartite subcubic graph, then 11𝑖(𝐺) ≤ 11𝑛0 (𝐺) + 7𝑛1 (𝐺) + 5𝑛2 (𝐺) + 4𝑛3 (𝐺) + 𝑏 1 (𝐺) + 2𝑏 2 (𝐺). The proof of Theorem 10.62 given in [109] uses vertex weighting arguments and assigns weights 4, 5, 7, and 11 to the vertices of degrees 3, 2, 1, and 0, respectively, in the subcubic graph 𝐺. They defined the 𝑡-weight of 𝐺 as the sum of the weights of the vertices of 𝐺 and the 𝑤-weight of 𝐺 as the sum of the 𝑡-weight and the terms 𝑏 1 (𝐺) + 2𝑏 2 (𝐺). Using the notion of a 𝑡-weight and 𝑤-weight, they proved the upper bound given in the statement of Theorem 10.62. In the special case of Theorem 10.62 when 𝐺 is a connected cubic graph, we have the following result, showing that Conjecture 10.58 is true if there is no subgraph isomorphic to 𝐾2,3 . Theorem 10.63 ([109]) If 𝐺 is a connected 𝐾2,3 -free bipartite cubic graph of 4 order 𝑛, then 𝑖(𝐺) ≤ 11 𝑛. As an immediate consequence of Theorem 10.63, we observe that Conjecture 10.58 is true if we forbid 4-cycles. Corollary 10.64 ([236]) If 𝐺 is a connected 𝐶4 -free bipartite cubic graph of 4 order 𝑛, then 𝑖(𝐺) ≤ 11 𝑛.

Chapter 10. Domination and Forbidden Subgraphs

314 Triangle-free Graphs

Independent domination in triangle-free graphs was investigated in 2008 by Haviland [401] and later in 2012 by Goddard and Lyle [360], who proved the following bounds on the independent domination number (where part (c) was also established in 2010 by Shiu et al. [674]). Theorem 10.65 ([360]) If 𝐺 is a triangle-free graph of order 𝑛, then the following hold: (a) There exist graphs 𝐺 with 𝑖(𝐺) = 𝑛 − O (𝑛). 3 (b) If 𝛿(𝐺) ≥ 20 𝑛, then 𝑖(𝐺) ≤ 12 𝑛 and this bound is tight for 𝛿(𝐺) ≤ 14 𝑛.  (c) If 𝛿(𝐺) ≥ 14 𝑛, then 𝑖 ≤ max 𝑛 − 2𝛿(𝐺), 𝛿(𝐺) and this bound is tight. Equality is obtained for graphs such as the following: take a path 𝑃4 : 𝑣 1 𝑣 2 𝑣 3 𝑣 4 and replace each 𝑣 𝑖 with an independent set 𝐴𝑖 with the same neighborhood, where | 𝐴1 | = | 𝐴4 | = 12 𝑛 − 𝛿 and | 𝐴2 | = | 𝐴3 | = 𝛿. Goddard and Lyle [360] constructed triangle-free graphs 𝐺 with 𝑖(𝐺) > 12 𝑛 for 1 all 0 < 𝛿(𝐺) < 10 𝑛 as follows: for a positive integer 𝛿, let 𝐺 𝛿 be obtained from the corona 𝐶5 ◦ 𝐾1 of a 5-cycle by replacing each endvertex by an independent set of size 15 𝑛 − 𝛿 and replacing each vertex of the 5-cycle by an independent set of size 𝛿. 1 Then, 𝑖(𝐺 𝛿 ) > 12 𝑛 for all 𝛿 < 10 𝑛. They posed the following question. Question 10.66 ([360]) Is it true that if 𝐺 is a triangle-free graph of order 𝑛 with 1 𝛿(𝐺) ≥ 10 𝑛, then 𝑖(𝐺) ≤ 12 𝑛?

10.3

Domination in Claw-free Graphs

Claw-free graphs are very well-studied in graph theory. The 1997 survey of claw-free graphs by Faudree et al. [271] remains a standard reference work on the topic. The class of claw-free graphs has important structural properties, and a full description is given by Chudnovsky and Seymour in a series of seven papers that appeared in the Journal of Combinatorial Theory (see for example [173]). For an overview, we refer the reader to their 2005 survey on the structure of claw-free graphs in [172]. In this section, we present selected results on the three core domination parameters in claw-free graphs.

10.3.1

Domination and Independent Domination Numbers

In 1978 Allan and Laskar [15] wrote the first paper on domination in claw-free graphs, in which they proved the well-known result that the domination number is equal to the independent domination number for all claw-free graphs. Theorem 10.67 ([15]) If 𝐺 is a claw-free graph, then 𝛾(𝐺) = 𝑖(𝐺). Proof Let 𝐺 be a claw-free graph. Among all 𝛾-sets of 𝐺, let 𝑆 be chosen so that the subgraph 𝐺 [𝑆] induced by 𝑆 contains the fewest edges. We show that 𝑆 is an independent set. Suppose, to the contrary, that there exist vertices 𝑢 and 𝑣

Section 10.3. Domination in Claw-free Graphs

315

in 𝑆 that are adjacent. Since 𝑆 is a minimal dominating set, it follows by Lemma 2.72 that ipn[𝑣, 𝑆] ≠ ∅ or epn[𝑣, 𝑆] ≠ ∅. Since 𝑢 and 𝑣 are adjacent vertices, we note that ipn[𝑣, 𝑆] = ∅, implying that the 𝑆-external private neighborhood of 𝑣 is nonempty, that is, epn[𝑣, 𝑆] ≠ ∅. If there are two (distinct) vertices 𝑣 1 and 𝑣 2 in epn[𝑣, 𝑆] that are not adjacent, then the subgraph 𝐺 [{𝑢, 𝑣, 𝑣 1 , 𝑣 2 }] is a claw in 𝐺, a contradiction. Hence, the set epn[𝑣, 𝑆] forms a clique, implying that if we replace an arbitrary vertex 𝑣 ′ from the set epn[𝑣, 𝑆], the resulting set the vertex 𝑣 in
𝑆 with ′ ′ 𝑆 = 𝑆 \ {𝑣} ∪ {𝑣 } is a 𝛾-set of 𝐺. However, 𝐺 [𝑆 ′ ] contains fewer edges than 𝐺 [𝑆], a contradiction. Therefore, 𝑆 is an independent set and hence is an ID-set of 𝐺, and so 𝑖(𝐺) ≤ |𝑆| = 𝛾(𝐺). By definition, 𝛾(𝐺) ≤ 𝑖(𝐺). Consequently, 𝛾(𝐺) = 𝑖(𝐺). Surprisingly, it was another 20 years before domination in claw-free graphs was studied in more depth. In 1998 Dutton and Brigham [252] observed the following simple structure of minimal dominating sets in claw-free graphs. Lemma 10.68 ([252]) The subgraph induced by any minimal dominating set in a claw-free graph is the disjoint union of complete graphs. Proof Let 𝑆 be an arbitrary minimal dominating set in a claw-free graph 𝐺. Suppose, to the contrary, that the subgraph 𝐺 [𝑆] contains an induced path 𝑣 1 𝑣 2 𝑣 3 on three vertices. Since ipn[𝑣 2 , 𝑆] = ∅, Lemma 2.72 implies that epn[𝑣 2 , 𝑆] ≠ ∅. If 𝑣 4 is an arbitrary vertex in epn[𝑣 2 , 𝑆], then the subgraph 𝐺 [{𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 }] is a claw in 𝐺, a contradiction. Hence, there is no induced path on three vertices in the subgraph 𝐺 [𝑆]. Equivalently, 𝐺 [𝑆] is the disjoint union of complete graphs. We note that if 𝐺 is the corona 𝐾 𝑘 ◦ 𝐾1 of a complete graph 𝐾 𝑘 of order 𝑘 ≥ 1, then 𝐺 is a connected claw-free graph of order 𝑛 = 2𝑘 ≥ 2 satisfying 𝛾(𝐺) = 12 𝑛. Hence, if 𝐺 is a connected claw-free graph of order 𝑛 ≥ 2, then the upper bound in Theorem 6.2 of 𝛾(𝐺) ≤ 12 𝑛 cannot be improved. If 𝐺 is a claw-free graph of order 𝑛 with minimum degree at least 2, then the upper bound in Theorem 6.18 of 𝛾(𝐺) ≤ 25 𝑛 cannot be improved. For example, for 𝑘 ≥ 1 an integer, let 𝐺 be obtained from the disjoint union of 𝑘 5-cycles as follows. In each 5-cycle, select one vertex and designate it as a joining vertex, and add an
edge between its two neighbors in the cycle. Thereafter, add all 𝑘2 edges between the 𝑘 joining vertices so that they form a clique. The resulting connected claw-free graph 𝐺 has order 𝑛 = 5𝑘 and satisfies 𝛾(𝐺) = 25 𝑛. For example, when 𝑘 = 4 the resulting graph 𝐺 is illustrated in Figure 10.23, where the highlighted vertices form a 𝛾-set of 𝐺. We observe that of the seven graphs that belong to the family Bdom (see Figure 10.2), only the graphs 𝐵1 = 𝐶4 and 𝐵4 = 𝐶7 are claw-free. Hence, as a consequence of Theorem 6.18 and the above observations, we have the following result. Theorem 10.69 If 𝐺 is a connected claw-free graph of order 𝑛, then the following hold: (a) If 𝛿(𝐺) ≥ 1, then 𝛾(𝐺) ≤ 12 𝑛, and this bound is tight. (b) If 𝛿(𝐺) ≥ 2 and 𝐺 ∉ {𝐶4 , 𝐶7 }, then 𝛾(𝐺) ≤ 25 𝑛, and this bound is tight.

316

Chapter 10. Domination and Forbidden Subgraphs

Figure 10.23 A connected claw-free graph 𝐺 of order 𝑛 with 𝛾(𝐺) = 25 𝑛

Recall that Reed’s result in Theorem 6.20 states that if 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾(𝐺) ≤ 38 𝑛. Further, this bound is tight, as illustrated in Figure 6.10. In contrast to the result of Theorem 10.69, the tight 38 -bound in Theorem 6.20 on the domination number of a graph with minimum degree at least 3 can be improved to a 1 3 -bound in the class of claw-free graphs. In 2021 Babikir and Henning [45] proved the following result. Theorem 10.70 ([45]) If every vertex of a graph 𝐺 of order 𝑛 belongs to a triangle, then 𝛾(𝐺) ≤ 13 𝑛. If 𝐺 is a claw-free graph with minimum degree at least 3, then every vertex of 𝐺 belongs to a triangle. Hence, as an immediate consequence of Theorem 10.70, we have the following result. Theorem 10.71 ([45]) If 𝐺 is a claw-free graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾(𝐺) ≤ 13 𝑛. The upper bound in Theorem 10.71 is tight, even if the maximum degree is arbitrarily large. To construct such a family, we first recall that by a 3-prism we mean the graph 𝐶3 □ 𝐾2 shown in Figure 10.24.

Figure 10.24 The 3-prism 𝐶3 □ 𝐾2 Let (𝐶3 □ 𝐾2 ) − be the graph obtained from the 3-prism 𝐶3 □ 𝐾2 by deleting one edge that does not belong to a triangle. We call the two vertices of degree 2 in (𝐶3 □ 𝐾2 ) − the gluing vertices of (𝐶3 □ 𝐾2 ) − . For 𝑘 ≥ 2, let 𝐿 𝑘 be obtained from 𝑘 disjoint copies of (𝐶3 □
𝐾2 ) − by selecting one gluing vertex from each copy of (𝐶3 □ 𝐾2 ) − and adding all 𝑘2 edges joining these 𝑘 gluing vertices, and thereafter
adding all 𝑘2 edges joining the remaining 𝑘 gluing vertices. We note that 𝐿 𝑘 is a connected claw-free graph on 𝑛 = 6𝑘 vertices with 𝛿(𝐺) = 3 and Δ(𝐺) = 𝑘 + 1.

Section 10.3. Domination in Claw-free Graphs

317

When 𝑘 = 4, the graph 𝐿 4 is illustrated in Figure 10.25, where the gluing vertices are highlighted. Let Lclaw-free be the family of all such graphs 𝐿 𝑘 , where 𝑘 ≥ 2. We note that there exist graphs in Lclaw-free with arbitrarily large maximum degree.

Figure 10.25 A graph 𝐿 4 in the family Lclaw-free

Let 𝐺 ∈ Lclaw-free have order 𝑛 = 6𝑘, and so 𝐺 = 𝐿 𝑘 for some 𝑘 ≥ 2. Every dominating set of 𝐺 contains at least two vertices from each of the 𝑘 copies of (𝐶3 □ 𝐾2 ) − used to construct the graph 𝐺, implying that 𝛾(𝐺) ≥ 2𝑘. Conversely, selecting two vertices from each copy of (𝐶3 □ 𝐾2 ) − , one vertex from each triangle, produces a dominating set of 𝐺, and so 𝛾(𝐺) ≤ 2𝑘. Consequently, 𝛾(𝐺) = 2𝑘 = 13 𝑛. Hence, we have the following observation, showing that the upper bound in Theorem 10.71 is tight for claw-free graphs with minimum degree 3 and arbitrarily large maximum degree. Proposition 10.72 ([43]) If 𝐺 ∈ Lclaw-free has order 𝑛, then 𝛾(𝐺) = 𝑖(𝐺) = 𝛾t (𝐺) = 13 𝑛. As a special case of Theorem 10.71, if 𝐺 is a claw-free cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 13 𝑛, thereby improving the upper bound of 𝛾(𝐺) ≤ 38 𝑛 given in Theorem 6.23 for general cubic graphs. The connected claw-free cubic graphs achieving equality in this bound were characterized by Babikir and Henning [43]. For 𝑘 ≥ 1 an integer, the connected claw-free cubic graph 𝑁Δ,𝑘 is constructed in [43] as follows. Take 2𝑘 disjoint copies 𝑇1 , 𝑇2 , . . . , 𝑇2𝑘 of a triangle 𝐾3 , where 𝑉 (𝑇𝑖 ) = {𝑎 𝑖 , 𝑏 𝑖 , 𝑐 𝑖 }, and add the edges 𝑏 2𝑖−1 𝑏 2𝑖 and 𝑐 2𝑖−1 𝑐 2𝑖 for all 𝑖 ∈ [𝑘]. Further, add the edges 𝑎 2𝑖 𝑎 2𝑖+1 for all 𝑖 ∈ [𝑘], where addition is taken modulo 2𝑘, and so 𝑎 2𝑘 𝑎 2𝑘+1 is the edge 𝑎 2𝑘 𝑎 1 . Following the terminology in [43], we call the resulting graph 𝑁Δ,𝑘 a triangle-necklace with 2𝑘 triangles. We note that the triangle-necklace 𝑁Δ,1 , shown in Figure 10.26(a), is the 3-prism 𝐶3 □ 𝐾2 shown in Figure 10.24. The triangle-necklace 𝑁Δ,3 is shown in Figure 10.26(b). Let Ntriangle = {𝑁Δ,𝑘 : 𝑘 ≥ 1}. The graphs that achieve equality in the upper bound 𝛾(𝐺) ≤ 13 𝑛 for connected claw-free cubic graphs 𝐺 are precisely the graphs in the family Ntriangle . Theorem 10.73 ([43]) If 𝐺 is a connected claw-free cubic graph of order 𝑛, then 𝛾(𝐺) ≤ 13 𝑛, with equality if and only if 𝐺 ∈ Ntriangle .

Chapter 10. Domination and Forbidden Subgraphs

318

(a) 𝑁Δ,1

(b) 𝑁Δ,3

Figure 10.26 The triangle-necklaces 𝑁Δ,1 and 𝑁Δ,3

10.3.2

Total Domination Number

In 1998 Dutton and Brigham [252] observed the following simple structure of minimal total dominating sets in claw-free graphs. Lemma 10.74 ([252]) Every component in the subgraph induced by a minimal TD-set in a claw-free graph is constructed from a clique of order at least 2 by adding at most one pendant edge to each vertex of the clique. Proof Let 𝑆 be an arbitrary minimal TD-set in a claw-free graph 𝐺. Suppose that the subgraph 𝐺 [𝑆] contains an induced path 𝑣 1 𝑣 2 𝑣 3 on three vertices. By Lemma 4.25, we have |epn(𝑣 2 , 𝑆)| ≥ 1 or |ipn(𝑣 2 , 𝑆)| ≥ 1. If epn(𝑣 2 , 𝑆) ≠ ∅ and 𝑣 4 is an arbitrary vertex in epn(𝑣 2 , 𝑆), then the subgraph 𝐺 [{𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 }] is a claw in 𝐺, a contradiction. Hence, epn(𝑣 2 , 𝑆) = ∅, implying that |ipn(𝑣 2 , 𝑆)| ≥ 1. Hence, the vertex 𝑣 2 has a neighbor in 𝑆 of degree 1 that is uniquely totally dominated by 𝑣 2 but by no other vertex of 𝑆. Renaming vertices if necessary, we may assume that 𝑣 1 has degree 1 in 𝐺 [𝑆]. Let 𝐶 be the set of neighbors of 𝑣 2 in 𝐺 [𝑆] different from 𝑣 1 , together with the vertex 𝑣 2 , and so {𝑣 2 , 𝑣 3 } ⊆ 𝐶 = N𝐺 [𝑆 ] (𝑣 2 ) \ {𝑣 1 }. If there are two (distinct) vertices 𝑢 and 𝑤 in 𝐶 that are not adjacent, then the subgraph 𝐺 [{𝑢, 𝑤, 𝑣 1 , 𝑣 2 }] is a claw in 𝐺, a contradiction. Hence, the set 𝐶 forms a clique. Since 𝐺 is claw-free, the vertex 𝑣 2 has at most two neighbors of degree 1 in 𝐺 [𝑆]. Further, if 𝑣 2 has two neighbors of degree 1 in 𝐺 [𝑆], then 𝐶 = {𝑣 2 , 𝑣 3 } and the component of 𝐺 [𝑆] containing 𝑣 2 is a path 𝑃3 . Hence, we may assume that 𝑣 1 is the only neighbor of 𝑣 2 of degree 1 in 𝐺 [𝑆], for otherwise we have the desired structure of the component. Suppose that a vertex 𝑣 ∈ 𝐶 different from 𝑣 2 has a neighbor not in 𝐶. Similar arguments for the vertex 𝑣 2 ∈ 𝐶 show that such a neighbor of 𝑣 is unique and is a vertex of degree 1, and that every other neighbor of 𝑣 belongs to the clique 𝐶. This completes the proof of Lemma 10.74. We note that if 𝐺 is the 2-corona 𝐾 𝑘 ◦ 𝑃2 of a complete graph 𝐾 𝑘 of order 𝑘 ≥ 1, then 𝐺 is a connected claw-free graph of order 𝑛 = 3𝑘 ≥ 3 satisfying 𝛾t (𝐺) = 23 𝑛. Hence, if 𝐺 is a connected claw-free graph of order 𝑛 ≥ 2, then the upper bound in Theorem 4.27 of 𝛾t (𝐺) ≤ 23 𝑛 cannot be improved. Theorem 10.75 ([182]) If 𝐺 is a connected claw-free graph of order 𝑛 ≥ 3, then 𝛾t (𝐺) ≤ 23 𝑛, and this bound is tight. In contrast to the result of Theorem 10.75, the tight 47 -bound in Theorem 6.46 on the total domination number of a graph with minimum degree at least 2 can be

Section 10.3. Domination in Claw-free Graphs

319

improved to a 21 -bound in the class of claw-free graphs, if we exclude a family of graphs R ★ that we describe below. claw-free An elementary 4-subdivision of a nonempty graph 𝐺 is a graph obtained from 𝐺 by subdividing some edge four times. A 4-subdivision of 𝐺 is a graph obtained from 𝐺 by a sequence of zero or more elementary 4-subdivisions of edges of 𝐺. Adopting the notation in [281], we define a good edge of 𝐺 as an edge 𝑢𝑣 such that both N[𝑢] and N[𝑣] induce a clique in 𝐺 − 𝑢𝑣. A good 4-subdivision of 𝐺 is a 4-subdivision of 𝐺 obtained by a sequence of elementary 4-subdivisions of good edges (at each stage in the resulting graph). Let 𝑅1 , 𝑅2 , . . . , 𝑅7 be the seven graphs shown in Figure 10.27.

𝑅1

𝑅2

𝑅5

𝑅3

𝑅6

𝑅4

𝑅7

Figure 10.27 The seven graphs 𝑅1 , 𝑅2 , . . . , 𝑅7  Let R ★ 𝑖 = 𝐺 : 𝐺 is a good 4-subdivision of the graph 𝑅𝑖 for 𝑖 ∈ [7] , and let R★ be the family defined by claw-free R★ claw-free =

7 Ø

R★ 𝑖 .

𝑖=1

In 2008 Favaron and Henning [281] presented the following 12 -bound result on the total domination number of a connected claw-free graph with minimum degree at least 2 that improves the 47 -bound in Theorem 6.46 for general connected graphs with minimum degree at least 2. Theorem 10.76 ([281]) If 𝐺 is a connected claw-free graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then one of the following holds: (a) 𝛾t (𝐺) ≤ 12 𝑛. , in which case 𝛾t (𝐺) = 12 (𝑛 + 1). (b) The graph 𝐺 is an odd cycle or 𝐺 ∈ R ★ claw-free (c) The graph 𝐺 = 𝐶𝑛 where 𝑛 ≡ 2 (mod 4), in which case 𝛾t (𝐺) = 12 (𝑛 + 2). As an immediate consequence of Theorem 10.76, we have the following result. Corollary 10.77 ([281]) If 𝐺 is a connected claw-free graph of order 𝑛 with 𝛿(𝐺) ≥ 2, then 𝛾t (𝐺) ≤ 12 (𝑛 + 2), with equality if and only if 𝐺 is a cycle of length congruent to 2 modulo 4.

320

Chapter 10. Domination and Forbidden Subgraphs

We consider next claw-free cubic graphs. Let 𝑁2 be the claw-free cubic graph, called a diamond-necklace with two diamonds, shown in Figure 10.28. If 𝐺 = 𝐾4 , then 𝐺 has order 𝑛 = 4 and 𝛾t (𝐺) = 2 = 21 𝑛. If 𝐺 = 𝑁2 , then 𝐺 has order 𝑛 = 8 and 𝛾t (𝐺) = 4 = 12 𝑛.

Figure 10.28 A diamond-necklace 𝑁2 with two diamonds Recall that Archdeacon et al. [35] showed in Theorem 6.52 that if 𝐺 is a graph of order 𝑛 with 𝛿(𝐺) ≥ 3, then 𝛾t (𝐺) ≤ 12 𝑛. Further, recall that the infinite family of connected graphs that achieve equality in this bound are all cubic graphs and are characterized in Theorem 6.59. In 2004 Favaron and Henning [279] proved that the 1 2 -bound also holds for claw-free cubic graphs, but showed that in this case, equality only holds for two small graphs, namely 𝐾4 and 𝑁2 . Theorem 10.78 ([279]) If 𝐺 is a connected claw-free cubic graph of order 𝑛, then 𝛾t (𝐺) ≤ 12 𝑛, with equality if and only if 𝐺 ∈ {𝐾4 , 𝑁2 }. In 2010 Southey and Henning [684] showed that if we exclude the two graphs 𝐾4 and 𝑁2 , then the result of Theorem 10.78 can be strengthened. Theorem 10.79 ([684]) If 𝐺 ∉ {𝑁1 , 𝑁2 } is a connected claw-free cubic graph of order 𝑛, then 𝛾t (𝐺) ≤ 49 𝑛. In 2011 Lichiardopol [564] showed that equality in the bound of Theorem 10.79 is achieved if and only if 𝐺 ∈ {𝐺 18.1 , 𝐺 18.2 }, where 𝐺 18.1 and 𝐺 18.2 are the two claw-free cubic graphs of order 𝑛 = 18 shown in Figure 10.29(a) and (b), respectively. In particular, if 𝐺 ∈ {𝐺 18.1 , 𝐺 18.2 }, then 𝛾t (𝐺) = 8 = 49 𝑛, where the highlighted vertices in these two graphs are examples of 𝛾t -sets. Let F = {𝐾4 , 𝑁2 , 𝐺 18.1 , 𝐺 18.2 }. In 2021 Babikir and Henning [44] showed that if we exclude the four graphs in the family F , then the result of Theorem 10.79 can be improved. Theorem 10.80 ([44]) If 𝐺 ∉ F is a connected claw-free cubic graph of order 𝑛, then 𝛾t (𝐺) ≤ 37 𝑛. The bound in Theorem 10.80 is best possible, as may be seen by considering the graphs 𝐺 28.1 and 𝐺 28.2 , shown in Figure 10.30(a) and (b), respectively, constructed in [44]. If 𝐺 ∈ {𝐺 28.1 , 𝐺 28.2 }, then 𝐺 has order 𝑛 = 28 and 𝛾t (𝐺) = 12 = 37 𝑛, where the highlighted vertices of 𝐺 in the figure form a 𝛾t -set of cardinality 12. It remains, however, an open problem to characterize the graphs achieving equality in the upper bound of Theorem 10.80. Let 𝐺 30 and 𝐺 48 be the two claw-free cubic graphs shown in Figure 10.31(a) and (b), respectively. If 𝐺 = 𝐺 30 , then 𝐺 has order 𝑛 = 30 and 𝛾t (𝐺) = 12 = 25 𝑛,

Section 10.3. Domination in Claw-free Graphs

321

(a) 𝐺 18.1

(b) 𝐺 18.2

Figure 10.29 The graphs 𝐺 18.1 and 𝐺 18.2

(a) 𝐺 28.1

(b) 𝐺 28.2

Figure 10.30 The graphs 𝐺 28.1 and 𝐺 28.2

where the highlighted vertices in Figure 10.31(a) form a 𝛾t -set of 𝐺 of cardinality 12. If 𝐺 = 𝐺 48 , then 𝐺 has order 𝑛 = 48 and 𝛾t (𝐺) = 18 = 38 𝑛, where the highlighted vertices in Figure 10.31(b) form a 𝛾t -set of 𝐺 of cardinality 18.

(a) 𝐺 30

(b) 𝐺 48

Figure 10.31 The graphs 𝐺 30 and 𝐺 48

322

Chapter 10. Domination and Forbidden Subgraphs

Using matching results in cubic graphs, Favaron and Henning [280] in 2008 obtained the following upper bounds on the total domination number of a connected cubic graph that is claw-free and diamond-free. Theorem 10.81 ([280]) If 𝐺 is a connected (𝐾1,3 , 𝐾4 − 𝑒)-free cubic graph of order 𝑛 ≥ 6, then 𝛾t (𝐺) ≤ 52 𝑛, with equality if and only if 𝐺 = 𝐺 30 . Further, the authors in [280] obtained the following upper bound on the total domination of a connected cubic graph that is claw-free, diamond-free, and 𝐶4 -free. Theorem 10.82 ([280]) If 𝐺 is a connected (𝐾1,3 , 𝐾4 − 𝑒, 𝐶4 )-free cubic graph of order 𝑛 ≥ 6, then 𝛾t (𝐺) ≤ 38 𝑛, with equality if and only if 𝐺 = 𝐺 48 . It remains an open problem to determine a tight upper bound on the total domination number of a connected claw-free graph of sufficiently large order 𝑛 with 𝛿(𝐺) ≥ 3. Let 𝐹claw-free be the claw-free graph of order 7 shown in Figure 10.32.

Figure 10.32 The graph 𝐹claw-free For 𝑘 ≥ 2 an integer, let 𝐺 be obtained from the disjoint union of 𝑘 copies of
the graph 𝐹claw-free by adding all 𝑘2 edges between the 𝑘 vertices of degree 2 in each copy of 𝐹claw-free so that they form a clique. The resulting connected claw-free graph 𝐹𝑘 has order 𝑛 = 7𝑘 and satisfies 𝛾t (𝐹𝑘 ) = 3𝑘 = 37 𝑛. For example, when 𝑘 = 4 the resulting graph 𝐹4 is illustrated in Figure 10.33. Let Fclaw-free be the family of all such graphs 𝐹𝑘 , for 𝑘 ≥ 2.

Figure 10.33 A graph 𝐹4 in the family Fclaw-free Let 𝐺 ∈ Fclaw-free have order 𝑛, and so 𝐺 = 𝐹𝑘 for some 𝑘 ≥ 2. Every TD-set of 𝐺 contains at least three vertices from each of the 𝑘 copies of the graph 𝐹claw-free used to construct the graph 𝐺, implying that 𝛾t (𝐺) ≥ 3𝑘. Conversely, 𝛾t (𝐹claw-free ) = 3, and choosing a 𝛾t -set from each of the 𝑘 copies of the graph 𝐹claw-free in the graph 𝐺

Section 10.4. Summary

323

constructs a TD-set of 𝐺 of cardinality 3𝑘, and so 𝛾t (𝐺) ≤ 3𝑘. Consequently, 𝛾t (𝐺) = 3𝑘 = 73 𝑛. Hence, we have the following. Proposition 10.83 If 𝐺 ∈ Fclaw-free has order 𝑛, then 𝛾t (𝐺) = 37 𝑛. 𝑛 denote the family of all connected claw-free graphs of order 𝑛 with Let F≥3 𝛿(𝐺) ≥ 3. We note that the family Fclaw-free contains connected claw-free graphs with 𝛿(𝐺) = 3 having arbitrarily large order and maximum degree. Hence, by Proposition 10.83, ! 𝛾t (𝐺) 3 lim sup ≥ . 𝑛→∞ 𝐺 ∈ F 𝑛 𝑛 7 ≥3

It remains, however, an open problem to determine the limit of this supremum.

10.4

Summary

In this chapter, we presented improved bounds on the three core domination numbers of a graph with given structural restrictions, such as forbidding certain cycles or claws. We showed in Corollary 10.11 that if we forbid 4- and 5-cycles, then the upper bound on the domination number of a connected graph of order 𝑛 ≥ 14 with 𝛿(𝐺) ≥ 2 can be improved from the 25 -bound due to McCuaig-Shepherd to a 38 -bound, and this bound is tight. We showed in Theorem 10.37 that if we forbid induced 6-cycles, then the upper bound on the total domination number of a graph with minimum 6 degree at least 2 can be improved from the 47 -bound in Theorem 6.46 to a 11 -bound. We established bounds on the domination and total domination numbers of a graph in terms of their order and girth. We discussed several outstanding conjectures, foremost of which are the Verstraëte 13 -conjecture, namely Conjecture 10.23, and the Kostochka’s inspired 13 -conjecture, namely Conjecture 10.24, for domination in cubic graphs. In Theorem 10.71 we showed that Reed’s tight 38 -bound in Theorem 6.20 on the domination number of a graph with minimum degree at least 3 can be improved to a 1 3 -bound in the class of claw-free graphs. In the case of claw-free cubic graphs, the graphs that achieve equality in this 13 -bound are characterized in Theorem 10.73. In Theorem 10.76 we showed that the 47 -bound in Theorem 6.46 on the total domination number of a general connected graphs with minimum degree at least 2 can be improved to a 12 -bound. In Theorem 10.80 we showed that if 𝐺 is a connected claw-free cubic graph of order 𝑛, and 𝐺 is not one of four exceptional graphs, then 𝛾t (𝐺) ≤ 37 𝑛, and this bound is best possible.

Chapter 11

Domination in Planar Graphs 11.1

Introduction

In this chapter, we present results on the core domination parameters in planar graphs. We shall adopt the following terminology for planar graphs. A graph is said to be planar if it can be drawn in the plane in such a way that no two edges intersect, except at a vertex to which they are both incident. A planar graph so drawn in the plane is said to be embedded in the plane and is called a plane graph. An embedding of a plane graph divides the plane into regions called faces. More formally, the faces of a plane graph 𝐺 are the connected pieces of the plane that remain after the points in the plane that correspond to the vertices and edges of 𝐺 are removed. Every plane graph contains exactly one unbounded face, called the outer face; the other faces are the inner faces. The boundary of a face 𝐹 (including the outer face) is the subgraph induced by the vertices and edges incident with the face 𝐹. Vertices and edges incident to the outer face of a plane graph 𝐺 are called external vertices and external edges, respectively, of 𝐺. Vertices of 𝐺 that are not external vertices are called internal vertices, and edges of 𝐺 that are not external edges are called internal edges of 𝐺. Two faces are adjacent if they have at least one edge in common. A facial triangle of a plane graph 𝐺 is a triangle whose interior is a face of 𝐺. A face bounded by a triangle is called a triangular face. For example, the plane graph 𝐺 in Figure 11.1 has four faces 𝑓𝑖 for 𝑖 ∈ [4], where 𝑓4 is its outer face and 𝑓3 is a triangular face. 𝑓4 𝑓1

𝑓2

𝑓3

Figure 11.1 A plane graph with four faces A triangulated disc (also called a near-triangulation in the literature) is a (simple) 2-connected plane graph all of whose faces are triangles, except possibly the outer © Springer Nature Switzerland AG 2023 T. W. Haynes et al., Domination in Graphs: Core Concepts, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-031-09496-5_11

325

326

Chapter 11. Domination in Planar Graphs

face. A triangulated disc in which all faces, including the outer face, are triangles and any two face boundaries intersect in a single edge, a single vertex, or not at all is called a planar triangulation. A weak triangulated disc (also called a weak near-triangulation in the literature) is a plane graph all of whose faces are triangles, except possibly the outer face, and every vertex of 𝐺 is contained in a triangle. We remark that a weak triangulated disc may be disconnected. Moreover, every vertex of a weak triangulated disc is contained in a facial triangle. An outerplanar graph is a planar graph that has an embedding in the plane such that all vertices are external vertices. A planar (respectively, outerplanar) graph is maximal if 𝐺 + 𝑢𝑣 is not planar (respectively, outerplanar) for every two nonadjacent vertices 𝑢 and 𝑣 of 𝐺. Thus, a maximal outerplanar graph 𝐺, abbreviated mop in the literature, is a triangulated disc where every vertex of 𝐺 is an external vertex. We note that a planar triangulation is a maximal planar graph.

11.2 Domination in Planar Graphs In this section, we present results on domination in planar graphs. Grid graphs are a well-known class of planar graphs, but since they are covered in Chapter 17, we do not cover them here. Given a graph 𝐺 and an integer 𝑘, the DOMINATING SET problem is to decide if 𝐺 has a dominating set of cardinality at most 𝑘. As discussed in Chapter 3, the DOMINATING SET problem is a core NP-complete problem in combinatorial optimization and graph theory [325]. If the problem is restricted to planar graphs, it is known as the PLANAR DOMINATING SET problem, which remains NP-hard even when restricted to planar graphs of maximum degree three [325]. Hence, it is of interest to determine upper bounds on the domination number of a planar graph.

11.2.1

Domination in Planar Triangulations

An outward numbering of a triangulated disc 𝐺 of order 𝑛 is a numbering of the vertices from 1 to 𝑛 such that for every 𝑖 ∈ [𝑛], the set of vertices numbered 1 through 𝑖 induces a triangulated disc 𝐺 𝑖 for which each vertex of 𝑉 (𝐺) \ 𝑉 (𝐺 𝑖 ) is an external vertex of 𝐺 𝑖 , that is, lies on the outer face of 𝐺 𝑖 . In 1996 Matheson and Tarjan [585] studied dominating sets in triangulated discs and proved a tight upper bound on their domination numbers. Their key result is the following lemma, which shows that there is an outward numbering of a triangulated disc starting from any given triangle. Lemma 11.1 ([585]) Given an arbitrary triangle in a triangulated disc 𝐺, there is an outward numbering of 𝐺 that numbers the vertices of the given triangle 1, 2, and 3, in any desired order. Proof We proceed by induction on the order 𝑛 of a triangulated disc 𝐺 with a given triangle. We number the vertices of the given triangle arbitrarily 1, 2, and 3. For the

Section 11.2. Domination in Planar Graphs

327

inductive hypothesis, suppose that there is a numbering of the vertices 1, 2, . . . , 𝑖 for some 𝑖, where 3 ≤ 𝑖 < 𝑛, such that the vertex set [𝑖] induces a triangulated disk 𝐺 𝑖 and 1, 2, . . . , 𝑖 is an outward numbering of 𝐺 𝑖 . We describe how to extend this to an outward numbering of the vertices 1, 2, . . . , 𝑖 + 1. The triangulated disc 𝐺 properly contains the triangulated disc 𝐺 𝑖 and by definition, all vertices in 𝑉 (𝐺) \𝑉 (𝐺 𝑖 ) are external vertices of 𝐺 𝑖 . Since a triangulated disc is 2-connected, there exists a triangle 𝑇 of 𝐺 containing exactly two vertices of 𝐺 𝑖 . Thus, one vertex of 𝑇 is currently unnumbered (with no number assigned to it from [𝑖]), and two vertices of 𝑇 are numbered from [𝑖]. Let 𝑆 be the set of all such unnumbered vertices of 𝐺 that belong to a triangle with exactly one unnumbered vertex. For a vertex 𝑣 ∈ 𝑆, let {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } be the set of vertices of 𝐺 𝑖 adjacent to 𝑣, indexed clockwise around the boundary of the outer face of 𝐺 𝑖 , with 𝑣 1 chosen so that none of the external edges of 𝐺 𝑖 from 𝑣 1 clockwise to 𝑣 𝑘 is an external edge of 𝐺. We note that this naming of the neighbors of 𝑣 is possible since the external edges of 𝐺 𝑖 are partitioned by 𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 into 𝑘 parts, at most one of which can intersect the boundary of the outer face of 𝐺. We define 𝑇 (𝑣) as the set consisting of all triangles inside the cycle 𝐶𝑣 : 𝑣 𝑣 1 𝑣 2 . . . 𝑣 𝑘 𝑣, and so all vertices of such triangles belong to the cycle 𝐶𝑣 or to the interior region of the cycle 𝐶𝑣 . We define the score of the vertex 𝑣 to be the number of triangles in the set 𝑇 (𝑣). Among all vertices in the set 𝑆, we choose a vertex 𝑣 of minimum score. If there is some vertex of 𝑆 inside the cycle 𝐶𝑣 , then such a vertex would have a smaller score than 𝑣, contradicting our choice of the vertex 𝑣. Hence, the set 𝑇 (𝑣) must consist  exactly of the set {𝑣, 𝑣 1 , 𝑣 2 }, {𝑣, 𝑣 2 , 𝑣 3 }, . . . , {𝑣, 𝑣 𝑘−1 , 𝑣 𝑘 } , implying that the vertex 𝑣 can be numbered 𝑖 + 1. Recall that the domatic number dom(𝐺) is the maximum order of a partition of 𝑉 (𝐺) into dominating sets. We are now in a position to present the labeling argument given by Matheson and Tarjan [585] that the vertex set of a triangulated disc can be partitioned into three dominating sets. Theorem 11.2 ([585]) If 𝐺 is a triangulated disc, then dom(𝐺) ≥ 3. Proof By Lemma 11.1, there is an outward numbering of the vertices in the triangulated disc 𝐺 that starts from any given triangle. We describe next a labeling of the vertices with labels 𝑎, 𝑏, and 𝑐 such that the set of vertices with the same label is a dominating set of 𝐺. We label the vertices (using the labels 𝑎, 𝑏, and 𝑐) sequentially, taking care to preserve the following two properties: P1. The labeling constructed thus far yields three dominating sets of the subgraph induced by the current labeled vertices. P2. The boundary of the outer face of the triangulated disc defined by the vertices labeled so far has no two adjacent vertices with the same label. Initially, we label the vertices 1, 2, and 3 in the outward numbering of the vertices of 𝐺, with the labels 𝑎, 𝑏, and 𝑐, respectively. Since these three vertices form a triangle, properties P1 and P2 are true. In general, suppose that the vertices numbered 1, 2, . . . , 𝑖 are labeled, each with one of the labels 𝑎, 𝑏, and 𝑐, so that properties P1

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and P2 hold for some 𝑖, where 3 ≤ 𝑖 < 𝑛. Let 𝐺 𝑖 be the triangulated disc associated with the outward numbering 1, 2, . . . , 𝑖. We now consider the vertex 𝑣 numbered 𝑖 + 1 in the outward numbering of 𝐺. As in the proof of Lemma 11.1, the set of vertices in 𝐺 𝑖 to which vertex 𝑣 is adjacent in 𝐺 is a set {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } of vertices occurring consecutively on the boundary of the outer face of 𝐺 𝑖 where 𝑘 ≥ 2, with the vertices indexed clockwise around the boundary. The boundary of the outer face of 𝐺 𝑖+1 is formed from the boundary of the outer face of 𝐺 𝑖 by deleting the path 𝑣 1 𝑣 2 . . . 𝑣 𝑘 and replacing it with the path 𝑣 1 𝑣 𝑣 𝑘 . To label the vertex 𝑣, we consider two cases. Suppose that 𝑣 1 and 𝑣 𝑘 are labeled differently in 𝐺 𝑖 (with labels from the set {𝑎, 𝑏, 𝑐}). In this case, we label the vertex 𝑣 with the label not used for 𝑣 1 or 𝑣 𝑘 . This preserves properties P1 and P2. Suppose next that 𝑣 1 and 𝑣 𝑘 are labeled the same in 𝐺 𝑖 (with labels from the set {𝑎, 𝑏, 𝑐}). Property P2 implies that in this case, the vertices 𝑣 1 and 𝑣 𝑘 are not adjacent, and therefore 𝑘 ≥ 3 and the vertex 𝑣 2 has a different label than 𝑣 1 . In this case, we label the vertex 𝑣 with the label not used for 𝑣 1 or 𝑣 2 . Once again, this preserves both properties P1 and P2. Upon completion of the labeling of the vertices (using the labels 𝑎, 𝑏, and 𝑐), by property P1 the set of vertices with the same label is a dominating set of 𝐺 and so dom(𝐺) ≥ 3. The following upper bound on the domination number of a triangulated disc is an immediate consequence of Theorem 11.2. Corollary 11.3 ([585]) If 𝐺 is a triangulated disc of order 𝑛, then 𝛾(𝐺) ≤ 13 𝑛. Matheson and Tarjan [585] gave a linear algorithm for finding a dominating set of cardinality at most 𝑛3 in a triangulated disc, by modifying slightly the construction of the outward numbering in the proof of Lemma 11.1. That the bound of Corollary 11.3 is tight may be seen as follows. For 𝑘 ≥ 1, take 2𝑘 vertex-disjoint copies 𝑇1 , 𝑇2 , . . . , 𝑇2𝑘 of a triangle, where 𝑉 (𝑇𝑖 ) = {𝑢 𝑖 , 𝑣 𝑖 , 𝑤 𝑖 }, and add the edges 𝑢 2𝑖−1 𝑢 2𝑖 , 𝑢 2𝑖−1 𝑤 2𝑖 , and 𝑤 2𝑖−1 𝑤 2𝑖 for all 𝑖 ∈ [𝑘]. To complete the construction, for 𝑘 ≥ 2 add the edges 𝑤 2𝑖−1 𝑢 2𝑖+1 , 𝑤 2𝑖−1 𝑢 2𝑖+2 , and 𝑤 2𝑖 𝑢 2𝑖+2 for all 𝑖 ∈ [𝑘 − 1]. Let 𝐺 𝑘 denote the resulting triangulated disc of order 𝑛 = 6𝑘. The graph 𝐺 3 , for example, is illustrated in Figure 11.2. Let Gouterplanar = {𝐺 𝑘 : 𝑘 ≥ 1}.

Figure 11.2 The graph 𝐺 3 Each graph 𝐺 ∈ Gouterplanar of order 𝑛 is an outerplanar graph with 𝑛/3 vertices of degree 2. Every 𝛾-set of 𝐺 contains at least one vertex from the closed neighborhood of every vertex of degree 2. However, such sets form a partition of 𝑉 (𝐺), noting that

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no two vertices of degree 2 in 𝐺 have a common neighbor. Therefore, 𝛾(𝐺) ≥ 31 𝑛. The set consisting of the 𝑛/3 vertices of degree 2 is an independent dominating set of 𝐺, and so 𝛾(𝐺) ≤ 𝑖(𝐺) ≤ 13 𝑛. Consequently, 𝛾(𝐺) = 𝑖(𝐺) = 13 𝑛. In 2010 by Honjo et al. [500] generalized the result of Corollary 11.3 and showed that every triangulation on the projective plane, the torus, and the Klein bottle of order 𝑛 has a dominating set of cardinality at most 13 𝑛. They showed that the same conclusion holds for other surfaces. The reader is referred to [500] for more details and definitions. Corollary 11.3 given by Matheson and Tarjan [585] and its generalization given by Honjo et al. in [500] follow from a more general recent result (see, Theorem 10.70 in Chapter 10) that if every vertex of a graph 𝐺 of order 𝑛 belongs to a triangle, then 𝛾(𝐺) ≤ 13 𝑛. However, Honjo et al. [500] proved a stronger result, which generalizes Theorem 11.2. Theorem 11.4 ([500]) If 𝐺 is a triangulation on the sphere, the projective plane, the torus, or the Klein bottle, then dom(𝐺) ≥ 3. Moreover, this bound is best possible. Since the class of triangulated discs is a superclass of planar triangulations, as an immediate consequence of Corollary 11.3, we have the following result. Corollary 11.5 ([585]) If 𝐺 is a planar triangulation of order 𝑛, then 𝛾(𝐺) ≤ 13 𝑛. The bound in Corollary 11.5 is achieved, for example, by the planar triangulations 𝐾3 and the octahedron 𝐾2,2,2 , which we illustrate in Figure 11.3.

𝐾3

𝐾2,2,2

Figure 11.3 The planar triangulations 𝐾3 and the octahedron 𝐾2,2,2 A natural problem is to characterize the planar triangulations that achieve equality in the upper bound in Corollary 11.5. Moreover, if there are only finitely many graphs achieving equality in the bound, it would be interesting to determine if there is an improved upper bound when the order is sufficiently large, or whether the 13 -bound is asymptotically best possible. In 2020 Špacapan [688] made a breakthrough on these problems and questions when he showed that the triangle 𝐾3 and the octahedron 𝐾2,2,2 are the only planar triangulations achieving equality in the bound of Corollary 11.5. Moreover, he improved the upper bound in Corollary 11.5 for planar triangulations of order 𝑛 > 6. For this purpose, he used properties of weak triangulated discs, also referred to as

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weak near-triangulations in [688], to establish the improved bounds. Recall that a weak triangulated disc is a plane graph (possibly disconnected) all of whose faces are triangles, except possibly for the outer face, and every vertex of 𝐺 is contained in a triangle. By definition, every triangulated disc is a weak triangulated disc, but not conversely. An example of a weak triangulated disc in given in Figure 11.4.

Figure 11.4 A weak triangulated disc

The following selected properties of weak triangulated discs, given in [688], follow readily from the definition of a weak triangulated disc and properties of plane graphs. Recall that a block is a maximal nonseparable subgraph, or a maximal subgraph having no cut-vertices, and an endblock is a block containing at most one cut-vertex of 𝐺. Lemma 11.6 ([688]) If 𝐺 is a weak triangulated disc, then each of the following holds: (a) Every block of 𝐺 is either a triangulated disc or a copy of 𝐾2 . (b) Every endblock of 𝐺 is a triangulated disc. (c) The graph 𝐺 has a (vertex) 3-coloring such that each color class is a dominating set of 𝐺. As an immediate consequence of Lemma 11.6, we have the following bound on the domination number of a weak triangulated disc. Corollary 11.7 ([688]) If 𝐺 is a weak triangulated disc of order 𝑛, then 𝛾(𝐺) ≤ 13 𝑛. Špacapan [688] defined a weak triangulated disc 𝐺 to be reducible if there exists a set 𝐷 ⊆ 𝑉 (𝐺) and a vertex 𝑣 ∈ 𝐷 such that all of the following three properties hold: (a) 𝐷 ⊆ N𝐺 [𝑣], (b) |𝐷| ≥ 4, (c) 𝐺 − 𝐷 is a weak triangulated disc. A weak triangulated disc is irreducible if it is not reducible. The key result of Špacapan [688] is the following theorem. Theorem 11.8 ([688]) If a weak triangulated disc 𝐺 has a block that is not an outerplanar graph and does not have order 6, then 𝐺 is reducible. As a consequence of Corollary 11.7 and Theorem 11.8, we have the following result.

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Corollary 11.9 ([688]) If 𝐺 is a triangulated disc of order 𝑛 ≠ 6 that is not an outerplanar graph, then 𝛾(𝐺) < 31 𝑛. Proof Let 𝐺 be a triangulated disc of order 𝑛 ≠ 6 that is not an outerplanar graph. By Theorem 11.8, the graph 𝐺 is reducible. Therefore, there exists a set 𝐷 ⊆ 𝑉 (𝐺) and a vertex 𝑣 ∈ 𝐷 that dominates 𝐷 such that |𝐷| ≥ 4 and 𝐺
− 𝐷 is a weak triangulated disc. By Corollary 11.7, we have 𝛾(𝐺 − 𝐷) ≤ 13 𝑛 − |𝐷 | . Every dominating set of 𝐺 − 𝐷 can be extended to a dominating set of 𝐺 by adding the vertex 𝑣 to it and so
𝛾(𝐺) ≤ 𝛾(𝐺 − 𝐷) + 1 ≤ 13 𝑛 − |𝐷 | + 1 ≤ 13 (𝑛 − 4) + 1 < 13 𝑛. 1 3𝑛

As a consequence of Corollary 11.7 and Theorem 11.8, the upper bound 𝛾(𝐺) ≤ in Corollary 11.5 for planar triangulations of order 𝑛 can be improved if 𝑛 > 6.

Theorem 11.10 ([688]) If 𝐺 is a planar triangulation of order 𝑛 > 6, then 𝛾(𝐺) ≤ 17 53 𝑛. Proof Let 𝐺 be a planar triangulation of order 𝑛 > 6 and consider an embedding of 𝐺 in the plane. By definition, all faces (including the outer face) of 𝐺 are triangles. If 𝐺 is irreducible, then let 𝐺 ′ = 𝐺. If 𝐺 is reducible, then let 𝐺 ′ be obtained from 𝐺 by repeatedly applying Theorem 11.8 until the resulting graph 𝐺 ′ is irreducible. Thus, in this case, there exist vertex-disjoint sets 𝐷 1 , 𝐷 2 , . . . , 𝐷 𝑘 of vertices of 𝐺 such that 𝐺 1 = 𝐺 and where 𝑣 𝑖 ∈ 𝐷 𝑖 , 𝐷 ⊆ N𝐺 [𝑣 𝑖 ], |𝐷 𝑖 | ≥ 4, and 𝐺 𝑖+1 = 𝐺 𝑖 − 𝐷 𝑖 is a weak triangulated disc for all 𝑖 ∈ [𝑘]. Moreover, 𝐺 ′ = 𝐺 𝑘+1 = 𝐺 𝑘 − 𝐷 𝑘 and the graph 𝐺 ′ is irreducible. By Theorem 11.8, every block of 𝐺 ′ is outerplanar or has order 6. Let 𝐺 ′ have order 𝑛′ , and so 𝑛′ ≤ 𝑛 − 4𝑘. The set 𝐷 = {𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑘 } dominates all vertices in 𝐷 1 ∪ 𝐷 2 ∪ · · · ∪ 𝐷 𝑘 and has cardinality |𝐷 | = 𝑘 ≤ 14 (𝑛 − 𝑛′ ), implying that the vertices in 𝑉 (𝐺) \ 𝑉 (𝐺 ′ ) are dominated by a set of cardinality at most 1 ′ 4 (𝑛 − 𝑛 ). Claim 11.10.1 If 𝐺 ′ contains all three external vertices of 𝐺, then 𝛾(𝐺)
0, there exists a constant 𝑐 = 𝑐(𝑆, 𝑡, 𝜀), dependent on 𝑆, 𝑡 and 𝜀, such that if 𝐺 is a triangulation on 𝑆 with at most 𝑡 vertices of degree other than 6, then
𝛾(𝐺) ≤ 16 + 𝜀 𝑛 + 𝑐. Liu and Pelsmajer [567] proved Theorem 11.14 with the constant 𝑐 = 𝑐(𝑆, 𝑡, 𝜀)
given by 𝑐 = O (𝑔 3 + 𝑔𝑡 2 )/𝜀 , where 𝑔 is the genus of 𝑆. Since 𝑛 is a trivial upper bound for the domination number, as remarked in√[567], Theorem 11.14 is only of
interest when 𝑐(𝑆, 𝑡, 𝜀) < 𝑛, in which case 𝑡 = O 𝑛 .

11.2.2

Domination in Outerplanar Graphs

Recall that mop is the abbreviation for maximal outerplanar graph. The class of triangulated discs is a superclass of mops. Hence, the following corollary is an

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immediate consequence of Corollary 11.3. We note that this corollary appeared as far back as 1977 in Mitchell’s PhD dissertation [592, 594]. Corollary 11.15 ([585]) If 𝐺 is a mop of order 𝑛 ≥ 3, then 𝛾(𝐺) ≤ 31 𝑛. The family Gouterplanar = {𝐺 𝑘 : 𝑘 ≥ 1} constructed earlier (see Figure 11.2 for an illustration of a graph in the family) of triangulated discs are in fact mops. Hence, the upper bound for mops given in Corollary 11.15 is best possible. The first results on the topic of domination in outerplanar graphs date back to at least 1978 when Fisk [304] produced an elegant proof to the Art Gallery Problem, a celebrated problem in the field of computational geometry, that was posed by Victor Klee to Chvátal in 1973. Problem 11.16 (Art Gallery Problem) Determine the minimum number of guards that need to be placed in an art gallery so that each point in the interior of the gallery is within the line of sight of at least one guard. The Art Gallery Problem can be placed in a mathematical context by modeling the art gallery by a polygon representing its plane view. More formally, let 𝑛 ≥ 3 be an integer. A set 𝑆 of points within a simple plane polygon P𝑛 with 𝑛 vertices (or corner points) guards P𝑛 if, for every point 𝑝 in the interior of P𝑛 , there exists some point 𝑞 ∈ 𝑆 such that the straight line segment between 𝑝 and 𝑞 lies inside the polygon P𝑛 . In 1975 Chvátal [174] solved the Art Gallery Problem.   Theorem 11.17 (Chvátal’s Watchman Theorem [174]) For 𝑛 ≥ 3, 𝑛3 guards are always sufficient and sometimes necessary to guard a simple plane polygon with 𝑛 corner points. That the 13 -upper bound in the Chvátal’s Watchman Theorem 11.17 is tight may be seen as follows. Consider a comb-shaped art museum 𝑀𝑘 with 𝑛 = 3𝑘 walls. For example, the museum 𝑀5 is illustrated in Figure 11.6(a). For 𝑘 ≥ 2 and for each 𝑖 ∈ [𝑘], the point 𝑝 𝑖 illustrated in Figure 11.6(a) can only be seen by a guard positioned in the shaded triangle containing 𝑝 𝑖 as illustrated in Figure 11.6(b). Since these 𝑘 triangles are disjoint, at least 𝑘 guards are needed. Placing one guard on the leftmost point of the bottom line and one guard on the rightmost point of the bottom line, and placing one guard at each of the points 𝑝 2 , 𝑝 3 , . . . , 𝑝 𝑘−1 , yields a placement of 𝑘 guards so that each point in the interior of the gallery is within the line of sight of at least one guard. Therefore, 𝑛3 guards are sufficient and necessary to guard a comb-shaped museum 𝑀𝑘 with 𝑛 = 3𝑘 walls. 𝑝1

𝑝2

𝑝3

(a)

𝑝4

𝑝5

𝑝1

𝑝2

𝑝3

(b)

Figure 11.6 A comb-shaped museum 𝑀5

𝑝4

𝑝5

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In 1978 Fisk [304] provided a proof of the Chvátal’s Watchman Theorem. Fisk’s proof appeared in the celebrated “Proofs from The Book” by Aigner and Ziegler [10], reserved for only the most elegant mathematical proofs. We present Fisk’s proof below. Proof of Theorem 11.17 View the polygon P𝑛 of the art gallery as a graph whose vertices are the corner points of the polygon and whose edges are the sides of the polygon. We now triangulate the part of this (planar) graph corresponding to the interior of the polygon P𝑛 to form a new graph by repeatedly adding 𝑛−3 non-crossing edges on the inside of the polygon between nonadjacent vertices until all faces of the interior are bound by triangles. The resulting graph is a mop. To illustrate this construction of a mop from a polygon P𝑛 , consider for example the mop in Figure 11.7 constructed from the polygon given in Figure 11.6(a).

Figure 11.7 A mop constructed from the polygon 𝑀5 in Figure 11.6(a) We prove by induction on 𝑛 ≥ 3 that the resulting mop 𝐺 is 3-colorable, that is, we can color the vertices with three colors so that no two adjacent vertices of 𝐺 receive the same color. For 𝑛 = 3, the graph 𝐺 = 𝐾3 and the result is immediate. This establishes the base case. Let 𝑛 ≥ 4, and assume that every mop of order 𝑛′ , where 3 ≤ 𝑛′ < 𝑛, is 3-colorable. Let 𝐺 be a mop of order 𝑛 and let 𝑢𝑣 be an arbitrary edge added to the polygon P𝑛 when constructing 𝐺. The edge 𝑢𝑣 splits the graph into two smaller mops 𝐺 1 and 𝐺 2 both containing the edge 𝑢𝑣. By induction, we may color each graph 𝐺 1 and 𝐺 2 with three colors 1, 2, and 3. Renaming colors if necessary, we may choose color 1 for the vertex 𝑢 and color 2 for the vertex 𝑣 in each coloring. Combining these colorings yields a 3-coloring of the original graph 𝐺, as desired. For example, the mop in Figure 11.7 can be 3-colored with the colors 1 (red), 2 (blue), and 3 (green) as illustrated in Figure 11.8.

Figure 11.8 A 3-coloring of the mop in Figure 11.7

Every triangle in 𝐺 contains a vertex of each color 1, 2, and 3. Hence, each color class of 𝐺 is a dominating set of 𝐺. Since there are 𝑛 vertices, it follows by the Pigeonhole Principle that  at least one of the color classes, say the vertices colored 1, contains at most 𝑛3 vertices. We now place our guards at all vertices colored with color 1, thereby guarding the whole art gallery with at most 𝑛3 vertices.

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For example, each color class of the mop in Figure 11.8 has cardinality 𝑛/3 = 5, and so we can place our guards at any of the three color classes. We can therefore place our guards at all vertices colored with color 1 (red), as illustrated by the black vertices in Figure 11.9(a). The associated guards in the original polygon 𝑀5 in Figure 11.6 are given by the black points in Figure 11.9(b). This completes the proof of Theorem 11.17.

(a)

(b)

Figure 11.9 A placement of guards in the mop in Figure 11.7

Implicit in the above proof of Chvátal’s Watchman Theorem 11.17 by Fisk [304] is that the domination number for a mop of order 𝑛 ≥ 3 is at most 𝑛/3. Thus, the result of Corollary 11.15 was implicitly proved in 1978 by Fisk [304]. It also follows, trivially, that the independent domination number of a mop of order 𝑛 is at most 𝑛/3. In 2013 Campos and Wakabayashi [130] and independently Tokunaga [712] established an improved upper bound on the domination number of a mop by using fundamental properties of mops. Let 𝑓 be an inner face of a mop 𝐺 (that is embedded in the plane). The face 𝑓 is called an inner triangle if none of its edges are incident to the outer face. The following upper bound on the domination number of a mop is given in [130, 712]. Theorem 11.18 ([130, 712]) If 𝐺 is a mop of order 𝑛 ≥ 3 with 𝑘 ≥ 0 internal triangles, then 𝛾(𝐺) ≤ 14 (𝑛 + 𝑘 + 2). The authors in [130] first established results for mops without any internal triangles. Thereafter, they proved by induction the main result in Theorem 11.18 by utilizing chords which are edges of internal triangles. As observed in [130], if 𝐺 is a mop of order 𝑛 ≥ 4 having 𝑘 internal triangles, then 𝐺 has 𝑘 +2 vertices of degree 2.As an immediate consequence of Theorem 11.18 and our earlier observations, we have the following result. Corollary 11.19 ([130]) If 𝐺 is a mop of order 𝑛 ≥ 4 with 𝑛2 ≥ 2 vertices of degree 2, then 𝛾(𝐺) ≤ 14 (𝑛 + 𝑛2 ). The bound in Corollary 11.19 is tight. For example, consider the family Gouterplanar constructed earlier (and illustrated in Figure 11.2). If 𝐺 ∈ Gouterplanar , then 𝐺 = 𝐺 𝑘 for some 𝑘 ≥ 1 and 𝐺 has order 𝑛 = 6𝑘 with 𝑛2 = 2𝑘 vertices of degree 2. As observed earlier, 𝛾(𝐺) = 13 𝑛. Thus, 𝛾(𝐺) = 2𝑘 = 14 (𝑛 + 𝑛2 ). As observed in [242], every mop 𝐺 of order 𝑛 ≥ 4 has at least two but not more than 𝑛/2 vertices of degree 2, that is, if 𝐺 has 𝑛2 vertices of degree 2, then 2 ≤ 𝑛2 ≤ 12 𝑛.

Section 11.2. Domination in Planar Graphs

337

In 2017 Dorfling et al. [242] improved the upper bound in Corollary 11.19 in the case when 31 𝑛 < 𝑛2 ≤ 12 𝑛. Theorem 11.20 ([242]) If 𝐺 is a mop of order 𝑛 ≥ 4 with 𝑛2 vertices of degree 2 such that 𝑛2 > 13 𝑛, then    𝑛2 + 1   if 𝑛2 = 12 𝑛   2 𝛾(𝐺) ≤   𝑛 − 𝑛2  if 𝑛2 < 12 𝑛.  2 If 𝐺 ∈ Gouterplanar , then 𝐺 = 𝐺 𝑘 for some 𝑘 ≥ 1 and 𝐺 has order 𝑛 = 6𝑘 with 𝑛2 = 2𝑘 = 13 𝑛 vertices of degree 2. Thus, 𝛾(𝐺) = 13 𝑛 = 𝑛2 = 12 (𝑛 − 𝑛2 ). Adding triangles on some outer edges of 𝐺 that do not contain vertices of degree 2 shows tightness of the bound in Theorem 11.20 for all 𝑛2 with 13 𝑛 < 𝑛2 ≤ 12 𝑛. In 2016 Li et al. [561] improved the upper bound of Campos and Wakabayashi [130] and Tokunaga [712] given in Corollary 11.19. Their proof leverages the local structure of mops when vertices of degree 2 are at distance at least 3 on the outer cycle in the plane embedding of a mop. Theorem 11.21 ([561]) If 𝐺 is a mop of order 𝑛 ≥ 3 with ℓ pairs of consecutive vertices of degree 2 at distance at least 3 apart on the outer cycle, then    𝑛   if ℓ = 0    4 𝛾(𝐺) ≤  𝑛+ℓ   if ℓ ≥ 1.   4 That the bound in Theorem 11.21 is tight may once again be seen by taking a graph 𝐺 ∈ Gouterplanar . If 𝐺 = 𝐺 𝑘 for some 𝑘 ≥ 1, then 𝐺 has order 𝑛 = 6𝑘 with ℓ = 2𝑘 consecutive vertices of degree 2 at distance at least 3 on the outer cycle. Thus, 𝛾(𝐺) = 2𝑘 = 14 (𝑛 + ℓ). As a consequence of Theorem 11.21, the authors in [561] proved the following upper bound on the domination number for Hamiltonian maximal planar graphs. Theorem 11.22 ([561]) If 𝐺 is a Hamiltonian maximal planar graph of order 5 𝑛 ≥ 7, then 𝛾(𝐺) ≤ 16 𝑛. In 1931 Whitney [750] proved that every 4-connected maximal planar graph is Hamiltonian. Combining this result with Theorem 11.22, we have the following result. Corollary 11.23 ([561]) If 𝐺 is a 4-connected maximal planar graph of order 5 𝑛 ≥ 7, then 𝛾(𝐺) ≤ 16 𝑛.

11.2.3

Domination in Planar Graphs with Small Diameter

In 1996 MacGillivray and Seyffarth [577] initiated the study of domination in graphs of bounded diameter. They remarked that the restriction of bounding the diameter

338

Chapter 11. Domination in Planar Graphs

on the domination number of a planar graph is reasonable to impose because planar graphs with small diameter are often important in applications, as explained, for example, in the 1995 paper by Fellows et al. [289]. A tree with diameter 4 can have arbitrarily large domination number. The interesting question is what happens when the diameter of a planar graph is 2 or 3. MacGillivray and Seyffarth [577] proved that planar graphs with diameter 2 or 3 have bounded domination numbers. In particular, this implies that the domination number of such a graph can be determined in polynomial time. The domination number of (general) graphs with diameter 2 can be arbitrarily number large. For example, recall that we showed in Chapter 7 that the domination √︁ of a random √︁ graph 𝐺 ∈ G(𝑛, 𝑝) on 𝑛 vertices, where 𝑝 ≈ (2 ln(𝑛))/𝑛, is of the order 𝑛 ln(𝑛) (see Theorem 7.21 for a more precise statement). In contrast, MacGillivray and Seyffarth [577] established that the domination number of planar graph with diameter 2 is at most 3. This bound was subsequently improved in 2002 by Goddard and Henning [349]. To state their result, let 𝐺 9 be the planar graph with diameter 2 and domination number 3 shown in Figure 11.10.

Figure 11.10 The planar graph 𝐺 9 with diameter 2 and domination number 3

Theorem 11.24 ([349]) If 𝐺 is a planar graph with diam(𝐺) = 2, then 𝛾(𝐺) ≤ 2, except for the graph 𝐺 = 𝐺 9 of Figure 11.10 for which 𝛾(𝐺 9 ) = 3. Proof Sketch We present here a sketch of the proof given in [349]. Suppose that 𝐺 is a planar graph with diam(𝐺) = 2 and 𝛾(𝐺) > 2. Since a cut-set dominates a diameter-2 graph, the graph 𝐺 is 3-connected. Therefore, 𝐺 has an essentially unique embedding in the plane (see [381]). We fix such an embedding of 𝐺 in the plane. From the Jordan Closed Curve Theorem, a cycle 𝐶 in 𝐺 separates the plane into two regions, called the sides of 𝐶. Vertices on different sides of 𝐶 are separated by 𝐶. The side of 𝐶 that consists of the unbounded face is called the outside of 𝐶, while the side of 𝐶 that consists of the bounded region is called the inside of 𝐶. If there are vertices inside 𝐶 and vertices outside 𝐶, then 𝐶 is called a cut-cycle. The theorem is now proven by a series of lemmas. The key lemma is to establish the existence of a 4-cycle with special properties, namely the 4-cycle is not both induced and dominating, nor both non-induced and dominating, and therefore not dominating. Thereafter, it is shown that 𝐺 is isomorphic to the graph 𝐺 9 shown in Figure 11.10. Therefore, if 𝐺 is a planar graph with diameter 2, then either 𝛾(𝐺) = 2 or 𝐺 = 𝐺 9 , in which case 𝛾(𝐺) = 3. In their 1996 paper, MacGillivray and Seyffarth [577] proved that planar graphs with diameter 3 have bounded domination numbers.

Section 11.2. Domination in Planar Graphs

339

Theorem 11.25 ([577]) If 𝐺 is a planar graph with diam(𝐺) = 3, then 𝛾(𝐺) ≤ 10. The upper bound in Theorem 11.25 in the case when the order of the planar graph is sufficiently large was improved in 2002 by Goddard and Henning [349]. For this purpose, they first proved the following result in the case when the radius is 2. Theorem 11.26 ([349]) If 𝐺 is a planar graph with diam(𝐺) = 3 and rad(𝐺) = 2, then 𝛾(𝐺) ≤ 6. Proof Sketch We present here a sketch of the proof given in [349]. Let 𝐺 be a planar graph with diam(𝐺) = 3 and rad(𝐺) = 2, and fix an embedding of 𝐺 in the plane. We adopt the notation in the proof of Theorem 11.24. Moreover, we define a basic cycle as a cut-cycle with certain special properties. More precisely, a basic cycle is an induced cycle 𝑥 𝑣 1 𝑣 2 . . . 𝑣 𝑟 𝑥 such that on both sides of the cycle there is a vertex whose neighbors on the cycle are a subset of the two consecutive vertices farthest from 𝑥, specifically 𝑣 (𝑟 −1)/2 and 𝑣 (𝑟+1)/2 if 𝑟 is odd and 𝑣 𝑟/2 and 𝑣 𝑟/2+1 if 𝑟 is even. The strategy of the proof is to establish special properties of cut-cycles and to show the existence of a basic cycle of length at most 5. Thereafter, the theorem is established by a series of lemmas to bound the domination number when there exists such basic cycles (of short length). Goddard and Henning [349] proved that if 𝐺 is a planar graph of sufficiently large order with radius and diameter 3, then the maximum domination number of such a planar graph is at most one more than the maximum for radius 2 and diameter 3. Their key lemma is the following result. Lemma 11.27 ([349]) For a sufficiently large planar graph 𝐺 with rad(𝐺) = diam(𝐺) = 3, there exists a planar graph 𝐺 ′ with rad(𝐺 ′ ) ≤ 2, diam(𝐺 ′ ) ≤ 3, and 𝛾(𝐺) ≤ 𝛾(𝐺 ′ ) + 1. As an immediate consequence of Theorem 11.26 and Lemma 11.27, we have the result of Goddard and Henning [349] that if 𝐺 is a planar graph of sufficiently large order with diameter 3, then 𝛾(𝐺) ≤ 7. In 2006 Dorfling et al. [239] improved the result in Theorem 11.26 and showed that every planar graph with diameter 3 and radius 2 has total domination number (and therefore domination number) at most 5. We remark that their proof used the same approach as that used to prove Theorem 11.26 in [349], but with more detailed analysis and with the use of a computer. Theorem 11.28 ([239]) If 𝐺 is a planar graph with diam(𝐺) = 3 and rad(𝐺) = 2, then 𝛾t (𝐺) ≤ 5. In order to handle the case when 𝐺 is a planar graph with diameter 3 and radius 3, the authors in [349] proved that the maximum domination number of such a planar graph is at most four more than the maximum for radius 2 and diameter 3. This, together with the result of Theorem 11.28, yielded the following slight improvement on the MacGillivray-Seyffarth result given in Theorem 11.25.

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Theorem 11.29 ([239]) If 𝐺 is a planar graph with diam(𝐺) = 3, then 𝛾(𝐺) ≤ 9. As an immediate consequence of Lemma 11.27 and Theorem 11.28, we have the following improved upper bound on the domination number of a planar graph of sufficiently large order. Theorem 11.30 ([239]) Every sufficiently large planar graph with diameter 3 has domination number at most 6. MacGillivray and Seyffarth [577] gave the example illustrated in Figure 11.11 of a planar graph with diameter 3 and domination number 6, where the six highlighted vertices form a 𝛾-set of the graph. By duplicating the vertices of degree 2, their example can be used to construct planar graphs of arbitrarily large order with diameter 3 and domination number 6. Hence, the upper bound in Theorem 11.30 is tight.

Figure 11.11 A planar graph with diameter 3 and domination number 6

MacGillivray and Seyffarth [577] showed that if we restrict the graphs in the statement of Theorem 11.25 to the class of outerplanar graphs, then the upper bound on the domination number can be decreased to 3. Theorem 11.31 ([577]) If 𝐺 is an outerplanar graph with diam(𝐺) = 3, then 𝛾(𝐺) ≤ 3. Analogous results on bounds on the domination number for other surfaces were presented in [349]. Theorem 11.32 ([349]) For each surface, there are finitely many graphs with diameter 2 and domination number more than 2. Theorem 11.33 ([349]) For each orientable surface, there is a maximum domination number of graphs with diameter 3.

Section 11.3. Total Domination in Planar Graphs

11.3

341

Total Domination in Planar Graphs

In this section, we present selected results on total domination in outplanar graphs and planar graphs with small diameter.

11.3.1

Total Domination in Outerplanar Graphs

In 2017 Dorfling et al. [242] proved an analogous result to Corollary 11.19 for total domination in mops. As observed earlier, if 𝐺 is a mop of order 𝑛 ≥ 4 with 𝑛2 vertices of degree 2, then 2 ≤ 𝑛2 ≤ 21 𝑛. Theorem 11.34 ([242]) If 𝐺 is a mop of order 𝑛 ≥ 3 with 𝑛2 vertices of degree 2, then  2(𝑛 − 𝑛2 )   if 𝑛2 > 13 𝑛 and 𝑛 ≥ 5   3 𝛾t (𝐺) ≤ 𝑛 + 𝑛2   otherwise.   3 As shown in [242], both bounds in Theorem 11.34 hold, but which is the better bound depends on 𝑛2 , that is, if 𝐺 is a mop of order 𝑛 ≥ 3 with 𝑛2 vertices of degree 2, then 𝛾t (𝐺) ≤ min 23 (𝑛 − 𝑛2 ), 13 (𝑛 + 𝑛2 ) . We note that if 𝑛2 > 13 𝑛, then it follows that 23 (𝑛 − 𝑛2 ) < 13 (𝑛 + 𝑛2 ). Several constructions showing tightness of the bounds in Theorem 11.34 are provided in [242]. We briefly present one such construction in the case when 𝛾t (𝐺) = 23 (𝑛 − 𝑛2 ). The family Gouterplanar constructed in Section 11.2.1 can be modified to a family Houterplanar = {𝐻 𝑘 : 𝑘 ≥ 1}, where each graph 𝐻 𝑘 has order 𝑛 = 10𝑘 with 𝑛2 = 4𝑘 vertices of degree 2 and total domination number 𝛾t (𝐻 𝑘 ) = 4𝑘 = 23 (𝑛 − 𝑛2 ). We omit the details of the construction and rather show the graph 𝐻3 ∈ Houterplanar in Figure 11.12 to illustrate the general construction.

Figure 11.12 The graph 𝐻3 Recall the basic result of Corollary 11.15 that if 𝐺 is a mop of order 𝑛 ≥ 3, then 𝛾(𝐺) ≤ 13 𝑛. A natural question is to determine an analogous upper bound (that depends only on 𝑛) for the total domination number of mops. This question was explored in 2016 by Dorfling et al. [241]. Let 𝐺 12.1 and 𝐺 12.2 be the two mops shown in Figure 11.13. Theorem 11.35 ([241]) If 𝐺 is a mop of order 𝑛 ≥ 5 and 𝐺 ∉ {𝐺 12.1 , 𝐺 12.2 }, then 𝛾t (𝐺) ≤ 25 𝑛.

Chapter 11. Domination in Planar Graphs

342

(a) 𝐺 12.1

(b) 𝐺 12.2

Figure 11.13 The mops 𝐺 12.1 and 𝐺 12.2

That the bound of Theorem 11.35 is tight may be seen as follows. Let 𝐹 be a copy of the mop illustrated in Figure 11.14(a), where we designate the two highlighted vertices as link vertices of 𝐹. For 𝑘 ≥ 1, take 𝑘 vertex-disjoint copies of the mop 𝐹 and add edges between the link vertices in such a way as to construct a mop. Let 𝐹𝑘 denote the resulting graph (which is not unique). We call each of the 𝑘 copies of the graph 𝐹 used to build the graph 𝐹𝑘 a unit of 𝐹𝑘 . A graph 𝐹3 , for example, constructed in this way is illustrated in Figure 11.14(b). Let Fouterplanar consist of all such graphs 𝐹𝑘 , where 𝑘 ≥ 1, that can be constructed in this way. Let 𝐺 ∈ Fouterplanar have order 𝑛 and so 𝐺 = 𝐹𝑘 for some 𝑘 ≥ 1 and 𝑛 = 5𝑘. Every TD-set in 𝐺 must contain at least two vertices from each of the 𝑘 units in 𝐺, implying that 𝛾t (𝐺) ≥ 2𝑘. The set consisting of the 2𝑘 vertices of 𝐺 that are neither link vertices nor vertices of degree 2 in 𝐺 form a TD-set of 𝐺, and so 𝛾t (𝐺) ≤ 2𝑘. Consequently, 𝛾t (𝐺) = 2𝑘 = 25 𝑛.

(a) 𝐹

(b) 𝐹3

Figure 11.14 The mop 𝐹 used to build a mop 𝐹3 ∈ Fouterplanar We remark that the proof of Theorem 11.35 is quite different than the proof of Corollary 11.3. Recall that the proof given by Matheson and Tarjan [585] followed from labeling arguments (and also follows from the more general result given in Theorem 10.70 in Chapter 10). However, the proof of Theorem 11.35 relied on a detailed case analysis due in part to the existence the two exceptional graphs 𝐺 12.1 and 𝐺 12.2 shown in Figure 11.13. Subsequently, in 2017 Lemańska et al. [558] presented an alternate proof to that given in [241] and they discussed a relation between total domination in mops and the concept of watched guards in simple polygons. As observed by O’Rourke [620] and others, every mop has a unique Hamiltonian cycle. Following the notation of Lemańska et al. [558], we refer to an edge that belongs to the Hamiltonian cycle of a mop as a Hamiltonian edge and to every other edge of the mop as a diagonal.

Section 11.3. Total Domination in Planar Graphs

343

Key properties of mops used by Lemańska et al. [558] are the following results of O’Rourke [619]. Lemma 11.36 ([619]) If 𝐺 is a mop of order 𝑛 ≥ 10, then there exists a diagonal edge 𝑑 of 𝐺 such that 𝑑 partitions 𝐺 into two mops sharing the common edge 𝑑, one of which contains exactly 5, 6, 7, or 8 Hamiltonian edges of 𝐺. Lemma 11.37 ([619]) If 𝐺 is a mop of order 𝑛 ≥ 4 and 𝑒 is a Hamiltonian edge of 𝐺, then the graph resulting from contraction of the edge 𝑒 is a mop of order 𝑛 − 1. Lemańska et al. [558] used the special diagonal with the property in the statement of the Lemma 11.36, together with the edge contraction property of mops from Lemma 11.37 that allowed them to prove the desired result by induction on the order of a mop. An application of Theorem 11.35 is the following modification of the Art Gallery Problem 11.16, where now we require the additional condition that each guard is watched by at least one other guard. Problem 11.38 (Watched Art Gallery Problem) Determine the minimum number of guards that need to be placed in an art gallery so that the following hold: (a) Every point in the interior of the gallery is within the line of sight of at least one guard. (b) Every guard is within the line of sight of some other guard. Recall that for 𝑛 ≥ 3, a set of points 𝑆 within a simple plane polygon P𝑛 with 𝑛 corner points, guards P𝑛 if for every point 𝑝 in the interior of P𝑛 , there exists some point 𝑞 ∈ 𝑆 such that the straight line segment between 𝑝 and 𝑞 lies inside the polygon P𝑛 . If the set 𝑆 of points has the additional property that every point 𝑝 ∈ 𝑆 is seen by some other point 𝑞 ∈ 𝑆 (that is, the straight line segment between points 𝑝 and 𝑞 lies inside the polygon P𝑛 ), then we call the set 𝑆 a watched guard set in P𝑛 (called a guarded guard set in [590]). In 2003 Michael and Pinciu [590] solved Problem 11.38. set Theorem 11.39 ([590]) For 𝑛 ≥ 5, a minimum watched guard   in a simple plane polygon P𝑛 with 𝑛 corner points has cardinality at most 3𝑛−1 , and this bound is 7 best possible. The authors in [590] constructed a polygon 𝑃𝑛 with 𝑛 = 12 corner points for which a minimum watched guard set has cardinality exactly (3𝑛 − 1)/7 = 5. Triangulating the part of the (planar) graph corresponding to the interior of their polygon P12 (as explained in the proof of Theorem 11.17) produces one of the two mops 𝐺 12.1 and 𝐺 12.2 shown in Figure 11.13. As observed by Lemańska et al. [558], if we modify the Watched Art Gallery Problem 11.38 by relaxing property (a) to require that only the 𝑛 corner points of polygon P𝑛 be guarded, while maintaining property (b) (that every guard is watched by at least one other guard), then as an immediate consequence of Theorem 11.35, we have the following result.

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Corollary 11.40 ([558]) For 𝑛 ≥ 5, the vertices of a simple plane polygon P𝑛 with 𝑛 corner points can be guarded by at most 52 𝑛 watched guards, apart from some 12-vertex polygons that require five watched guards. In 2018 Alvarado et al. [21] presented a unified proof of the results of Corollary 11.15 and Theorem 11.35. For 𝑘 a positive integer, they defined a 𝑘-component dominating set of a graph as a dominating set 𝐷 of 𝐺 with the additional property that every component of the subgraph 𝐺 [𝐷] of 𝐺 induced by 𝐷 has order at least 𝑘. We denote the minimum cardinality of a 𝑘-component dominating set of 𝐺 by 𝛾 𝑘,comp (𝐺). We note that 𝛾(𝐺) = 𝛾1,comp (𝐺) and 𝛾t (𝐺) = 𝛾2,comp (𝐺). For every positive integer 𝑘, Alvarado et al. [21] constructed a set H𝑘 of graphs, each member of which has order at least 4𝑘 + 4 and at most 4𝑘 2 − 2𝑘. For 𝑘 = 1, we have 4𝑘 + 4 > 4𝑘 2 − 2𝑘, and so H1 is necessarily empty. Moreover, H2 consists exactly of the two exceptional mops 𝐺 12.1 and 𝐺 12.2 shown in Figure 11.13. The main result in [21] is the following. Theorem 11.41 ([21]) If 𝑘 and 𝑛 are positive integers with 𝑛 ≥ 2𝑘 + 1 and 𝐺 is a mop of order 𝑛, then l 𝑘𝑛 m    if 𝐺 ∈ H𝑘    2𝑘 + 1 𝛾 𝑘,comp (𝐺) ≤ j  𝑘𝑛 k   otherwise.   2𝑘 + 1 Since H1 = ∅, Theorem 11.41 implies Corollary 11.15. Furthermore, since H2 = {𝐺 12.1 , 𝐺 12.2 }, Theorem 11.41 implies Theorem 11.35.

11.3.2

Total Domination in Planar Graphs with Small Diameter

Recall that in Section 11.2.3, we discussed results on domination in planar graphs with small diameter. In this section, we present results on total domination in planar graphs with small diameter. One can readily deduce from Theorem 11.24 that the total domination number of a planar graph with diameter 2 is at most 3, as first observed in 2006 by Dorfling et al. [239]. Theorem 11.42 ([239]) If 𝐺 is a planar graph with diam(𝐺) = 2, then 𝛾t (𝐺) ≤ 3. Proof Let 𝐺 be a planar graph with diam(𝐺) = 2. If 𝐺 = 𝐺 9 , where 𝐺 9 is the graph of Figure 11.10, then 𝛾t (𝐺) = 3. If 𝐺 ≠ 𝐺 9 , then by Theorem 11.24, we have 𝛾(𝐺) ≤ 2. Let {𝑢, 𝑣} be a 𝛾-set in 𝐺. If 𝑢 and 𝑣 are adjacent, then {𝑢, 𝑣} is a TD-set in 𝐺, and so 𝛾t (𝐺) ≤ 2. If 𝑢 and 𝑣 are not adjacent, then since diam(𝐺) = 2, there is a common neighbor 𝑤 of 𝑢 and 𝑣. In this case, {𝑢, 𝑣, 𝑤} is a TD-set in 𝐺 and so 𝛾t (𝐺) ≤ 3. There are infinitely many planar graphs 𝐺 with diam(𝐺) = 2 and 𝛾t (𝐺) = 3. An infinite family of such graphs can be obtained, for example, from a 5-cycle 𝑣 1 𝑣 2 𝑣 3 𝑣 4 𝑣 5 𝑣 1 by replacing the vertices 𝑣 2 and 𝑣 5 with nonempty independent sets 𝑉2 and 𝑉5 , respectively, and adding all edges between vertices in 𝑉2 and {𝑣 1 , 𝑣 3 } and adding all

Section 11.3. Total Domination in Planar Graphs

345

Figure 11.15 A planar graph with diameter 2 and total domination number 3

edges between vertices in 𝑉5 and {𝑣 1 , 𝑣 4 }. As an illustration, a graph in such a family with |𝑉2 | = |𝑉5 | = 4 is shown in Figure 11.15. It remains and open problem to characterize planar graphs with diameter 2 and total domination number 3. In 2009 Henning and McCoy [471] obtained a characterization in the special case when the planar graphs with diameter 2 have certain structural properties. More specifically, they define a graph 𝐺 to satisfy the domination-cycle property if there is some 𝛾-set of 𝐺 that is not contained in any induced 5-cycle. The authors in [471] characterized the planar graphs with diameter 2 and total domination number 3 that satisfy the domination-cycle property and showed that there are exactly 34 such planar graphs. In 2020 Goddard and Henning [355] showed that if we restrict the graphs in the statement of Theorem 11.42 to the class of outerplanar graphs, then the upper bound on the total domination number can be improved slightly. Theorem 11.43 ([355]) If 𝐺 is an outerplanar graph with diam(𝐺) = 2, then 𝛾t (𝐺) = 2, unless 𝐺 = 𝐶5 . Proof Let 𝐺 be an outerplanar graph with diam(𝐺) = 2. If 𝐺 has a cut-vertex, then this vertex dominates the graph, and 𝛾(𝐺) = 1 and 𝛾t (𝐺) = 2. Hence, we may assume that 𝐺 is 2-connected. Thus, we can embed 𝐺 in the plane so that all vertices of 𝐺 are external vertices on a Hamiltonian cycle 𝐶 and all edges lie within 𝐶. If there is a chord, then the ends of the chord form a TD-set of 𝐺, since there must be path between vertices on either side of the chord, and so 𝛾t (𝐺) = 2. If there