Connectivity in Graphs 9781487584863

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CONNECTIVITY IN GRAPHS

MA THEM A TI CAL EXPOSITIONS Editorial Board

a L M. COXETE~ D. & DELURY G. F. D. DUFF, G. DE B. ROBINSON (Secretary), W. J. WEBBER

w.

T. SHARP

Volumes Published

No. 1. The Foundations of Geometry by G. DEB. ROBINSON No. 2. Non-Euclidean Geometry by H. s. M. COXETER No. 3. The Theory of Potential and Spherical Harmonics by w. J. STERNBERG and T. L. SMITH No. 4. The Variational Principles of Mechanics by CORNELIUS LANCZOS

No. 5. Tensor Calculus by J. L. SYNGE and A. E. SCHILD No. 6. The Theory of Functions of a Real Variable by R.

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No. 7. General Topology by w ACLA w SIERPINSKI (translated by c. CECILIA KRIEGER) No. 8. Bernstein Polynomials by G. G. LORENTZ No. 9. Partial Differential Equations by G. F. D . DUFF No. 10. Variational Methods for Eigenvalue Problems bys. H. GOULD

No. 11. Differential Geometry by ERWIN KREYSZIG No. 12. Representation Theory of the Symmetric Group by G. DE B. ROBINSON

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No. 14. Rings and Radicals by N. J. DIVINSKY No. 15. Connectivity in Graphs by w. T. TUTTE

MATHEMATICAL EXPOSITIONS No. 15

CONNECTIVITY IN GRAPHS by

W. T. TUITE

UNIVERSITY OF TORONTO PRESS LONDON: OXFORD UNIVERSITY PRESS

©

UNIVERSITY OF TORON10 PRESS

1966

WNDON : OXFORD UNIVERSITY PRESS

Reprinted in 2018 ISBN 978-1-4875-7296-9 (paper)

PRINTED IN GREAT BRITAIN BY WILLMER BROTHERS LIMITED BIRKENHEAD, ENGLAND

PREFACE GRAPH THEORY is now too extensive a subject for adequate presentation in a book of this size. Faced with the alternatives of writing a shallow survey of the greater part of graph theory or of giving a reasonably deep account of a small part, I have chosen the latter. I have written indeed as though the present book were to be the first section of a three-volume work. The book is concerned mainly with the theory of connection, non-separability, and triple connection in general graphs. Questions involving the symmetry of graphs are also studied. But important parts of graph theory, such as the study of planar graphs and that of the decomposition of a graph into factors of specified kinds, have been left aside for possible future treatment. It is thought that some of the results presented here have not been published before. These include the theory of vertices of attachment given in Chapter 1, the results on isthmus-avoiding paths in Chapter 5, the proof of the uniqueness of the 7-cage in Chapter 8, th~ theory of the decomposition of a 2-connected graph into 3-connected cleavage units given in Chapter 11, and the theory of nodal 3-connection in Chapter 12. I wish to thank Professor F. A. Scherk for his careful scrutiny of the original manuscript, resulting in the detection of a number of errors.

W. T. TUTIE

University of Waterloo July, 1965

V

CONTENTS V

PREFACE CHAPTER

I.I. 1.2. 1.3. I .4. 1.5. CHAPTER

2.1. 2.2. 2.3. 2.4. 2.5. 2.6. CHAPTER

3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3. 7.

1. GRAPHS AND SUBGRAPHS Graphs Valency Subgraphs Vertices of attachment The complement of a subgraph

3 4

5

6 8

2. CONNECTION MODULO A .SUBGRAPH

Maximal and minimal subgraphs Detachment modulo a subgraph Connection mod J Components mod J The case J = Q Representative graphs of subgraph-pairs 3. TREES, ARCS, AND POLYGONS Isthmuses The Betti numbers Trees Spanning trees Arcs Arcs and connection Polygons

9 9 IO 11 13 14

17 18 19

21 22

23 26

CHAPTER 4. PATHS

4.1. 4.2. 4.3. 4.4.

Paths Semi-simple paths Paths in arcs and polygons Distance

vii

28

29 30

32

viii

CONTENTS

CHAPTER

5.

EULER PATIIS

5.1. 5.2. 5.3. 5.4.

Preliminaries The construction of Euler paths Arbitrarily traceable graphs Isthmus-avoiding paths 5.5. The number of Euler circuits

CHAPTER

6.1. 6.2. 6.3. 6.4. 6.5. CHAPTER

7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7. CHAPTER

8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8. 7. 8.8. CHAPTER

9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9. 7.

6. MAPPINGS General mappings The inverse of a mapping Contractive mappings Isomorphisms The abstract graphs 7. AUTOMORPHISMS The group of a graph Examples Transitivity Advancing paths s-transitive graphs Girth s-regular graphs

8.

41

43 44 45

46 48

52 53 54

56 59 61

62

GIRTH

Distance Bipartite graphs Girth and distance Graphs of girth < 4 Cages Cages with a divalent exterior graph The 7-cage An existence theorem

9.

36 36 38 40

65 67 68 70 72 73

77 81

SEPARATION

Separable and non-separable graphs Classification of non-separable graphs Cyclic elements Cut-vertices Cut-forests A contractive mapping Cyclic connection

84 85 86 89 90 92

95

~

CONTENTS CHAPTER

10.1. 10.2. 10.3. 10.4. 10.5. 10.6. CHAPTER

I 1. 1. 11.2. 11.3. 11.4." 11.5. 11.6. CHAPTER

12.1. 12.2. 12.3. 12.4. 12.5. 12.6. 12.7. 12.8. INDEX

10. THREE-CONNECTION Connectivity Flags Connectivity and symmetry Operations on vertices and edges Essential edges The 3-connected graphs 11. THE STRUCTURE OF TWO-CONNECTED Preliminaries A mapping of subgraphs Hinges Cleavage Augmented graphs Cleavage units 12. NODAL THREE-CONNECTION Nodes and branches Nodal subgraphs Nodal n-connection Nodal 2-connection Homeomorphisms Adjunction of an arc Some constructions Subdivisions

97 99 101 103 104 111 GRAPHS

114 116 118 120 121 124

127 128 129 130 131 I 33 137 140 143

CONNECTIVITY IN GRAPHS

1 GRAPHS AND SUBGRAPHS 1.1. Graphs. A graph G consists of a set V(G) of elements called vertices and a set E(G) of elements called edges, together with a relation of incidence, which associates with each edge two vertices called its ends. The two ends of an edge need not be distinct. An edge with distinct ends is called a link, and an edge with coincident ends is a loop. It is not strictly necessary that V(G) and E(G) be disjoint sets. But if any element is used both as an edge and as a vertex its two roles must be carefully distinguished. The graph G is finite if both V( G) and E( G) are finite sets, and is infinite otherwise. In the present work we discuss only finite graphs. From now on, therefore, the word "graph" is always to mean a finite graph unless the contrary is explicitly·stated. We admit the case in which the sets V(G) and E(G) are both null. We then describe G as a null graph. A graph with just one vertex x and no edge is a vertex-graph. This vertex-graph is logically distinct from the vertex x and we denote it by [x]. A link-graph is a graph consisting of a single link and its two incident vertices. Similarly, a loop-graph consists of a single loop with its one incident vertex. The two ends of an edge A are said to be adjacent in G, or to be joined by A in G. Thus a vertex is adjacent to itself only if it is incident with a loop. G is called a clique if it has no loops and if each two distinct vertices of G are joined by exactly one edge. A clique with just n vertices is an n-clique. Thus the null graphs, vertex-graphs, and link-graphs are the 0-cliques, I-cliques, and 2-cliques respectively. It is customary to represent a graph by a diagram in which each vertex is shown as a dot and each edge is shown as a curve joining the two dots representing its ends. Two such curves may intersect at a point that does not represent a vertex. However, we shall always break one of the two curves at such a point, to suggest that it passes behind the other. Which of the two curves is broken does not matter. Examples of such diagrams are given below: Figure I.IA represents a 5-clique, and Figure 1.lB represents a graph with three vertices, one link, and three loops. 3

4

CONNECTIVITY IN GRAPHS

FIGURE

1.IA .

• FIGURE

1.1B.

Examples of graphs are provided by the convex polyhedra. Consider any convex polyhedron Q, for example a cube or regular dodecahedron. Then the edges and vertices of Q are the edges and vertices respectively of a graph. In this graph the vertices incident with any edge are its ends in the geometrical sense. A second graph associated with Q is perhaps less obvious : its vertices and edges are the faces and edges of Q respectively, and the ends of an edge A in the graph are the two faces of the polyhedron that have A as a 1;ide. 1.2. Valency. The number of vertices of a graph G is denoted by a.o(G), and the number of edges by a.1( G). Thus for an n-clique we have

a.o(G) = n,

a.i(G)

= ½n(n - 1).

The valency val (G, x) of a vertex x of G is the number of incident edges, loops being counted twice. A vertex of zero valency is said to be isolated. There is, for example, one isolated vertex in the graph of Figure 1. 1B. We note that in an

GRAPHS AND SUBGRAPHS

5

n-clique the valency of each vertex is n - 1, and that in a loop-graph the valency of the single vertex is 2. From these definitions we have at once 1.21.

L val(G, x) = 2a1(G) •

.xeV(G)

As a corollary to this theorem we have 1.22. In any graph the number of-vertices of odd valency is even. 1.3. Subgraphs. A graph His a subgraph of a graph G if V(H) s;; V(G), E(H) s;; E(G), and each edge of H has the same ends in Has in G. We then say that G contains H and write H s;; G. If, in addition, H is not identical with G, we write H c G and say that H is a proper subgraph of G, or is properly contained in G. If His a (proper) subgraph of G, we say also that G is a (proper) $Upergraph of H. For sets as well as subgraphs we use the symbol

to denote proper inclusion only. The two following results are immediate consequences of the definition of a subgraph. c

1.31. Any subgraph of a subgraph of G is a subgraph of G. 1.32. Suppose that V s;; V( G) and E s;; E(G). Then in order that there shall exist a subgraph Hof G such that V(H) = V and E(H) = E it is necessary and sufficient that V shall include the two ends in G of each member of E. The graph of Figure l.IB, for example, has one subgraph consisting of the three vertices and the single link, and another defined by the vertex of valency 5 and its two incident loops. SupposethatS s;; E(G). By l.32thereisasubgraphH ofGsuch that V(H) = V(G) and E(H) = S. His a spanning subgraph of G, denoted by G: S. There is another subgraph K of G such that E(K) = Sand V(K) is the set of all vertices of G that are incident in G with members of S. We call K the reduction of G to Sand denote it by G · S. Clearly a reduction has no isolated vertices. Moreover, G · Sis obtained from G : S by suppressing the isolated vertices of the latter subgraph. Now suppose X s;; V(G). By 1.32 there is a subgraph of G whose vertices are the members of X and whose edges are those edges of G that have both ends in X. We denote this subgraph by G[X] and call it the subgraph of G generated by X. Let Hi, H 2, ••• , H 11 be subgraphs of G. By 1.32 there is a subgraph Hof G such that V(H) = B

k

u V(H;),

i=l

k

E(H) = u E(H;). i=l

6

CONNECTIVITY IN GRAPHS

We call H the union of the subgraphs H 1 and write k

H = u H 1 = H 1 u Hau •.• u Hk. 1=1

Similarly, there is a subgraph K of G such that V(K)

k

= n

1=1

V(H1),

E(K) =

k

n

1=1

E(H1) .

We call Kthe intersection of the subgraphs H 1 and write K

=

k

n H 1 = H 1 n H 2 n . .. n H 1 •

1=1

The subgraphs Hi, H 2, •• • , Hk of Gare said to be disjoint if the intersection of any two distinct ones is the null subgraph of G.

1.4. Vertices of attachment. From now on we use the symbols 0 and n to denote a null set and a null graph respectively. We also write I SI for the number of members of a finite set S . Let H be a subgraph of a graph G. A vertex of attachment of Hin G is a vertex of H that is incident in G with an edge not belonging to E(H). We denote the set of vertices of attachment of Hin G by W(G, H). We note the formulae: 1.41. W(G, G)

=

W(G, Q)

= 0.

1.42. W(G, H) £ V(H). The less trivial properties of vertices of attachment depend on the following identity 1.43. In this and the remaining theorems of the present section the symbols Hand K denote arbitrary subgraphs of G. 1.43.

W(G, H)

=

W(K, H n K) u {V(H) n W(G, Hu K)}.

Proof. W(G, H) is the set of all vertices of H incident with edges of the set E(G) - E(H). But

= {E(G) - E(H u K)} u {E(K) - E(H n K)}. Moreover, W(K, H n K) is the set of all vertices of H incident with edges of E(K) - E(H n K), and V(H) n W(G, Hu K) is the set of all vertices of H E(G) - E(H)

incident with edges of E(G) - E(H u K). The theorem follows. 1.431. W(H u K, H) = W(K, H n K). l.432. If H £ K, then W(K, H) £ W(G, H).

7

GRAPHS AND SUBGRAPHS

The first of these corollaries is obtained by writing G = H u Kin 1.43 and using 1.41. To derive the second from 1.43 we have only to observe that in the case considered W(K, H n K) = W(K, H). 1.44. W(G, Hu K) u W(G, H

n

K)

= W(G, H) u W(G, K).

Proof. From 1.43 and the corresponding formula with Hand K interchanged we have V(H) n W(G, Hu K) s;; W(G, H), V(K) n W(G, Hu K) s;; W(G, K). Hence W(G, Hu K) s;; W(G, H) u W(G, K), by (1.42). Writing H n K for Hin 1.43, we obtain W(G, H

n

K)

= W(K, H n

s;;

u {V(H n K) n W(G, K)} by 1.431, W(H u K, H) u W(G, K), by 1.432. W(G, H) u W(G, K), K)

s;; On the other hand, W(G, H) s;; W(G, H n K) u W(G, Hu K) by 1.43 and 1.432, and we may interchange Hand Kin this formula. Combining these results, we obtain 1.44.

1.45. W(G, Hu K)

n

V(H

n

K) s;; W(G, H) n W(G, K) s;; W(G, H

n

K).

Proof. W(G, Hu K) n V(H n K)

= {W(G, Hu K) n V(H)} n {W(G, Hu K) n V(K)} s;; W(G, H) n W(G, K), by 1.43. Moreover, W(G, H) n W(G, K) s;; W(G, K) n V(H),

by 1.43 with H replaced by H

n

s;; W(G, H

n

by 1.42,

K),

K.

1.46. Let q(H, K) denote the number of vertices x of G such that x belongs to W(G, H) and W(G, K) but not to W(G, Hu K). Then

I W(G,H u K) I + I W(G,H () K) I = I W(G,H) I + I W(G,K) I - q(H,K). Proof. I W(G, Hu K) I + I W(G, H n K) I = I W(G, Hu K) u W(G, H n K)I + I W(G, Hu K) n W(G, H n K)I = I W(G, H) u W(G, K) I + I W(G, Hu K) (\ W(G, H) (\ W(G, K) I, by (1.44) and (1.45),

= I W(G, H) u W(G, K) I + I W(G, H) (\ W(G, K) I - q(H, K) = I W(G, H) I + I W(G, K) I - q(H, K).

8

CONNECTIVITY IN GRAPHS

1.S. The complement of a subgraph. Consider any subgraph Hof a graph G. By 1.32 there is a subgraph H of G whose edges are the members of E( G) -

E(H), and whose vertices are the members of V(G) - V(H) together with the vertices of attachment of Hin G. We call H the complement of Hin G. 1.51. The vertices of attachment of Hin Gare those vertices of attachment of Hin G that are not isolated in H. Proof. Since Hu H = G, we have W(G, H) = W(H, H n H) s;; V(H), by 1.431 and 1.42. Hence W(G, H) s;; W(G, H), by the definition of H. Conversely, suppose that u E W(G, H). Then u E V(H). Accordingly, u is a vertex of attachment of H in G if and only if it is not an isolated vertex of H. Theorem 1.51 has the following obvious consequences. 1.52. No vertex of attachment of H is isolated in H. Proof. Any such vertex is a vertex of attachment of H, by 1.51, and is therefore incident in G with an edge of H. 1.53. The complement R of H may be derived from H by deleting those members of W(G, H), if any, that are isolated in H. 1.54. The complement of R is H.

2 CONNECTION MODULO A SUBGRAPH 2.1. Maximal and minimal subgraphs. Consider any class Q of subgraphs of a given graph G. We say that a member Hof Q is maximal if it has no other member of Q as a supergraph. Similarly, His minimal if it has no other member of Q as a subgraph. 2.11. If H is any member of Q there exist maximal and minimal members, K and L respectively, of Q such that L s;;; H s;;; K.

Proof. If His not maximal, there exists H 1 E Q such that H c H 1• If H 1 is not maximal, there exists H 2 E Q such that H 1 c H 2, and so on. Since G is finite, the resulting chain of subgraphs must terminate in a maximal member K of Q such that H s;;; K . The existence of L may be established analogously.

As an example let J be a fixed subgraph of G, and let Q be the class of all subgraphs Hof G such that J u H = G. Then it is clear that the only minimal member of Q is the complement of J and that the only maximal member is G.

2.2. Detachment modulo a subgraph. In the remainder of this chapter we use the symbol J to denote a fixed subgraph of a given graph G. A subgraph H of G is said to be detached modulo J, or to be J-detached, if W(G, H) s;;; V(J). The expression "moduloJ" may be abbreviated as "modJ."

' ,,

,,

,,

, , ,,

'

t-B

FIGURE

9

2.2A.

10

CONNECTIVITY IN GRAPHS

Some examples of J-detached subgraphs are shown in Figure 2.2A. Here J is the subgraph of G, without isolated vertices, whose edges correspond to the broken lines. Four J-detached subgraphs, A, B, C, and D, are indicated. A is a vertex-graph, Bis a link-graph, Chas just three edges and three vertices, and D includes all the edges and vertices not belonging to A, B, C, or J. The union of any of A, B, C, and D with any subgraph of J is also J-detached. The fundamental properties of ]-detached subgraphs may be listed as follows. 2.21. G and the subgraphs ofJ are detached mod J. 2.22. If a subgraph H of G is J-detached, then H is J-detached. 2.23. Let Hand K be subgraphs of G. Then they are both detached mod J if and only if H

u

Kand H

n

Kare both detached mod J.

The first of these properties may be regarded as a consequence of 1.41 and 1.42. The second follows from 1.51, and the third from 1.44.

2.3. Connection mod J. We say that G is connected modulo J, or J-connected, if it has no J-detached subgraph other than G itself and the subgraphs of J. It is convenient to extend this definition as follows: we say that a subgraph H of G is connected modulo J, or J-connected, if it is connected modulo its subgraph H n J. 2.31. Let J be a proper subgraph ofG, and let G be connected mod J. Then E( J) =

0, and no isolated vertex of G belongs to V(J).

Proof. Since J c G, the complement J of J is not a subgraph of J. But it is detached mod J, by 2.21 and 2.22. Hence J = G, by hypothesis, and the theorem follows by the definition of a complement.

2.32. Let Hand K be J-connected subgraphs of G such that H graph ofJ. Then H u K is J-connected.

n

K is not a sub-

Proof. Suppose the contrary. Then there is a proper subgraph X of H u K that is detached mod (Hu K) n J in H u K and is not a subgraph of J. We may adjust the notation so that H n Xis not a subgraph of J. Now W(H, H n X) = W(H u X, X) £ W(H u K, X), by 1.431 and 1.432. Hence H n Xis detached mod H n Jin H. Since His J-connected, it follows that H n X = H, that is H £ X. We now have H n K £ X, so that Kn Xis not a subgraph of J. We may therefore interchange Hand Kin the preceding argument and obtain K £ X. Combining these results, we have X = H u K, contrary to the definition of X.

CONNECTION MODULO A SUBGRAPH

ll

The next two theorems are obvious from the definitions. 2.33. Let x be any vertex of G. Then [x] is J-connected. 2.34. Let A be any edge of G. Then G · {A} is J-connected. Theorem 2.32 can be used to determine J-connected subgraphs of G when G and J are given. The procedure is that used in the following proof. 2.35. Let H be a J-connected subgraph of G. Then we can find a subgraph K of G such that H ~ K and K is both J-connected and J-detached. Proof. If H is not J -detached, it has a vertex x of attachment that is not in V(J). We can find an edge A of G that is incident with x but not in E(H). By 2.32 and 2.34, Hu (G · {A}) is a J-connected subgraph of G having Has a proper subgraph. We now repeat this operation with Hu (G · {A}) replacing H, and so on. The process must terminate in a graph K having the required properties.

Consider, for example, the graph of Figure 2.2A. The subgraphs A and B are J-connected, by 2.33 and 2.34. Each link-graph in C is J-connected, and therefore C is J-connected, by two applications of 2.32. The subgraph D, however, is not J-connected, for one of its proper subgraphs is a loop-graph that is detached mod Jin D but is not contained in J. 2.4. Components mod J. Let us write P(G, J) for the class of all ]-detached subgraphs of G that are not contained in J, and Q(G, J) for the class of all J-connected subgraphs of G that are not contained in J. The two classes are related by the following theorem.

2.41. The subgraphs of G belonging to P(G, J) ("\ Q(G, J) are the minimal members of P(G, J) and the maximal members of Q(G, J) . Proof. Let H be any member of P( G, J). Let K be any subgraph of H that is not a subgraph of J. By 1.43 we have

W(G, K) = W(H, K)

u {V(K) ("\ W(G, H)}.

But W( G, H) ~ V(J) . Hence K is detached mod J ("\ Hin H if and only if it is detached mod Jin G. We deduce that His J-connected if and only if it is minimal in P(G, J) . Thus P(G, J) ("\ Q(G, J) is the set of all minimal members of P(G, J). Any member of Q(G,J)isasubgraphofsomememberof P(G,J) ("\ Q(G,J), by 2.35. Since no minimal member of P(G, J) can be a subgraph of another, it follows that P(G, J) ("\ Q(G, J) is the set of all maximal members of Q(G, J). A subgraph of G is called a component of G mod J, or a J-component of G,

12

CONNECTIVITY IN GRAPHS

if it is J-detached and J-connected but not a subgraph of J. Thus, by 2.41, the J-components of Gare the minimal members of P(G, J) and the maximal members of Q(G, J). 2.42. LetCbeanyJ-componentofG.ThenE(J)n E(C) no isolated vertex of C.

=

0andV(J)contains

Proof. Since C is connected modulo its proper subgraph C n J, this result is a consequence of 2.31.

2.43. Suppose J is a proper subgraph of G. Then G is connected mod J if and only ifit is itself aJ-component ofG. Proof. If G is a J-component of G, then G is connected mod Jby the definition of a J-component. Conversely, if G is connected mod J, it is a maximal member of Q(G, J) and therefore a J-component of G.

2.44. Let C be any J-component of G and let H be any J-detached subgraph of G. Then either C s;; Hor C n H s;; J. Proof. C n His J-detached by 2.23. If it is not a subgraph of J, it must be identical with C, since C is minimal in P(G, J). But in the latter case Cs;; H.

As a corollary we have 2.441. Let C and D be distinct J-components of G. Then C n D s;; J. 2.45. G is the union of J and the J-components of G. Proof. Denote this union by U. Suppose U =I= G. Then V is not a subgraph of J. But Vis J-detached, by 2.22. Hence, by 2.11, there is a J-component C of G such that C s;; V. We have also C s;; U, by the definition of U. But E(U) n E(V) = 0, and V(U) n V(V) = W(G, U) s;; V(J). Hence Cs;; Un V s;; J, which is contrary to the definition of a /-component. We deduce that in fact U = G.

2.46. A subgraph H of G is J-detached if and only if it is a subgraph of J or a union of a subgraph of J with one or more J-components of G. Proof. The sufficiency of the condition follows from 2.21 and 2.23. To prove its necessity, suppose His any /-detached subgraph of G. Let Ube the union of H n J and all ✓-components C of G such that C n His not a subgraph of J. Then H s;; U by 2.45, and U s;; H, by 2.44. Hence H = U.

An edge A or vertex x of G not belonging to J is an edge or vertex respectively of just one J-component of G, by 2.441 and 2.45. This J-component can be determined by the procedure of 2.35, starting with the /-connected subgraph [x] or G · {A} of G.

CONNECTION MODULO A SUBGRAPH

13

In some problems it is necessary to replace J by another subgraph of G. The following theorem may then be helpful. 2.41. Let H be a subgraph of G that is not a subgraph ofJ. Let K be a subgraph of G such that H () K s;; J. Suppose further that H is J-connected. Then H is K-connected. Proof. By 2.45 there is a component C of H mod H () K that is not a subgraph of J. Then W(H, C) s;; V(H () K) s;; V(J). Hence C is detached mod H () J in H. Since H is ]-connected, it follows that C = H . Hence H is K-connected, by 2.43. In the graph of Figure 2.2A the subgraphs A, B, and C are ]-components. D is the union of two ]-components of G, one of which is a loop-graph. 2.5. The case J = n. An important special case of the foregoing theory arises when J = n. We abbreviate the terms "0-detached," "0-connected," and "0-component" as "detached," "connected," and "component" respectively. We note that the detached subgraphs of a graph Gare those without vertices of attachment, and that G is connected if and only if it has no proper non-null detached subgraph. Thus null graphs are connected. Moreover, all vertexgraphs, link-graphs, and loop-graphs are connected by 2.33 and 2.34. The chief properties of the components of G may be summarized as follows.

2.51. The components of G are (by definition) those subgraphs of G that are non-null, detached in G, and connected. They are the minimal non-null detached subgraphs o/G, and its maximal non-null connected ones (by 2.41). 2.52. Any non-null connected subgraph of G is a subgraph of a component of G (by 2.11 and 2.51). 2.53. If G is non-null, its components are disjoint and have G as their union (by 2.441 and 2.45). 2.54. A non-null subgraph H of G is detached components of G (by 2.46).

if and only if it is a union of

The components of a given graph G can be determined by the method of 2.35. The graph of Figure 2.2A has just two components, A and its complement. The number of components of a graph G is denoted by p 0(G). 2.55. p 0(G) 2.56. p0( G)

> 0, with equality if and only if G = n.

< 1 if and only if G is connected.

The first of these results follows from 2.53 and the fact that a component is non-null. The second then follows from 2.43 and 2.53, together with the fact that null graphs are connected.

14

CONNECTIVITY IN GRAPHS

2.57. If G is edgeless, then po(G) =

rx0(G).

Proof. Since all subgraphs of Gare detached, its components are its minimal non-null subgraphs, by 2.51. These are the vertex-graphs contained in G.

2.6. Representative graphs of subgraph-pairs. Let H and K be subgraphs of a graph G. Let L 8 , LK, and M be disjoint sets in 1-1 correspondence with the sets of components of H, K, and H n K respectively. For each element x of L 8 , LK, or M we denote the corresponding component of H, K, or H n K respectively by fx. Each component of H n K is contained in just one component of H and just one component of K, by 2.52 and 2.53. Hence we can define a graph R = R(H, K) as follows: V(R) = L 8 u LK, E(R) = M, and the ends of any edge m of Rare the vertices x E L 8 and y ELK such that/m £ fx n fy. We call Ra representative graph of the subgraph-pair {H, K} of G. Let Ube any subgraph of R. We define JU, a subgraph of Hu K, as the union of those subgraphs fx of H u K such that x is an edge or vertex of U. ~

- - - - -I 1. We can write G = G(Q), where Q

= (a0 ,

Ai, ai, A 2,

•• • ,

Am, am)

is a circular path in G of length m, by 4.36. Let us make the convention that any suffix may be increased or diminished by any multiple of m without changing the edge or vertex to which it refers. Then we can distinguish two automorphisms p and u of G. The first increases each suffix by 1, thus transforming Q into one of its rotations. The second transforms Q into Q- 1 • Thus it replaces each suffix i by m - i for vertices and by m + l - i for edges. By 6.45 there are just 2m automorphisms of G. They map Q onto the 2m rotations of Q and Q- 1 , by 4.36. Hence they are the mappings piui, where 0 < i < m and O 1 and no link of G appears as two consecutive edge-terms in P. An advancing path of lengths is referred to briefly as ans-route. We note two simple consequences of these definitions. 7.41. Every non-degenerate semi-simple path is advancing. 7.42. The inverse of an s-route is an s-route. 7.43. Let P be an advancing path in a graph G, and let k be a positive integer. Then we can construct in Gan advancing path PR such that either s(R) = k or PR has a monovalent terminus. If G is a polygon, the first alternative must hold and R is uniquely determined by P and k. Proof Let x be the terminus of P, and Xits last edge-term. Ifval(G, x) > 1, we can find an edge A incident with x that is either a loop or a link distinct from X. Let y be the other end of A. Then P(x, A, y) is advancing. Repeating this construction sufficiently often, we obtain the required path PR. If P is a polygon there is just one choice for A at each stage in the above co.n struction. For x is then incident either with just one loop and no links or with no loop and just two links. The theorem follows .

7.44. Let G be a connected graph that is not a polygon. Let (x, A, y) be any 1-route in G. Then we can construct in Gan advancing path P of the form (x, A, y)T, where T either has a monovalent terminus or is of the form T 1(y, A, x).

Proof We may suppose A to be a link since otherwise we may take P = (x, A, x, A, x). Starting with (x, A,y) we carry out the construction of 7.43, withk > cx 0 (G). We terminate it as soon as a vertex is repeated or a monovalent vertex is reached. We may assume that in fact some vertex c is repeated, since in the other alternative the theorem is satisfied. Suppose first that c is not x. Then we have an advancing path of the form (x, A, y)Q 1 Q 2 , where Q 2 is non-degenerate and re-entrant, and no edge occurs both in (x, A, y)Q 1 and in Q 2 • We may therefore take P to be the advancing path (x, A, y)Q 1 Q 2 Q1 - 1(y, A, x). In the remaining case we have a circular path (x, A, y)Q 1• The corresponding subgraph, H say, of G is a polygon, by 4.32. Since G is connected and not a polygon, there is a vertex of attachment, z say, of H. Write Q1 = Q3 Q,,

57

AUTOMORPHISMS

where z is the terminus of Q3 and the origin of Q4 • There is an edge B of G that is incident with z and does not occur in (x, A, y)Q1 • Let its other end bet. (See Figure 7.4A.)

:x;

R,

FIGURE

7.4A.

We may suppose B to be a link. For otherwise we can put P = (x, A, y)Q 3 (z, B, z)Q 3 - 1(y, A, x). We now repeat the preceding arguments with (z, B, t) replacing (x, A, y). Three cases arise. First, we may obtain an advancing path (z, B, t)R with a monovalent terminus. We can then put P

=

(x, A, y)Qa(z,B, t)R.

Secondly, we may obtain an advancing path (z, B, t)R(t, B, z). In this case we put P

= (x,

A, y)Qa(z, B, t)R(t, B, z)Q 3 - 1(y, A, x).

In the remaining case we have a circular path (z, B, t)R. Let u be the first vertex-term of R that is in V(H). We write R = R 1 R2 , where u is the terminus of R1 and the origin of R 2• We also write Q = UU1, where u is the terminus of U and the origin of U1 • This is the case illustrated in Figure 7.4A, though it has been thought unnecessary to show R 2 or to indicate U1 • We put P

=

(x, A, y)Qa(z, B, t)R1

u- (y, A, x). 1

7.45. Let G bea connected graph that is not apolygon.Let(a, A, b) and(x, X,y) be I-routes in G. Then we can construct in G an advancing path P of the form (a, A, b)Q such that P either has a monovalent terminus or is also of the form R(x, X, y).

Proof. If(a, A, b) = (x, X, y), we putP = (a, A, b). If(a, A, b) = (y, X, x), the theorem is true by 7.44. We may therefore suppose that A#- X.

58

CONNECTIVITY IN GRAPHS

The subgraphs G · {A} and G · {X} may have a common vertex. If not, we can construct an arc Lin G, with ends u and v say, such that L n (G · {A}) = [u] and L n (G · {X}) = [v], by 3.65 with J = Q. In the second alternative we write Qi for the simple path from u to v in L. In the first we take Q1 to be a degenerate path (u) = (v), where u = vis a common vertex of G · {A} and G · {X}. If b = u, we have an advancing path (a, A, b)Q1 • If a = u, we can construct an advancing path (a, A, b)T, where T either has a monovalent terminus or is of the form Ti(b, A, a), by 7.44. In the former alternative we may put P = (a, A, b)T. In the latter we have an advancing path (a, A, b)TQ1 • In all the cases remaining we have an advancing path (a, A, b)TQi, where Tmay be degenerate. Ifv = x, we can putP = (a, A, b)TQ1(x, X,y). Ifv = y, we can, by 7.44, construct an advancing path (y, X, x)U, where U either has a monovalent terminus or is of the form U1(x, X, y). In either case we can put P = (a, A, b)TQi(y, X, x)U. 1.46. Let G be a connected graph that is not a polygon. Let P1 and P 2 be advancing paths in G. Then we can construct in G an advancing path P of the form P1 Q such that P either has a monovalent terminus or is also of the form RP2 • Proof Write P 1 = Qi(a, A, b) and P 2 = (x, X, y)Q 2 • By 7.45 we can construct an advancing path T of the form (a, A, b)U such that T either has a monovalent terminus or is also of the form Ui(x, X, y). In the former alternative we write P = Q1 T, and in the latter P = Q1 TQ 2 •

Consider an s-route P

= (a0 ,

A 1 , ai, ... , As, as)

in a graph G. Let As+i be an edge, incident with as, that is either a loop or a link distinct from As. Let its other end be as+i• Then is another s-route in G. We call Q a successor of P. As a consequence of this definition, we have 7.47. Let P be ans-route in a graph G. Let p be the number of distinct edges of G incident with the terminus ofP. Then the number ofdistinct successors ofPis p or p - 1, according as the last edge-term of Pis a loop or a link. 7.48. Let P 1 and P 2 bes-routes in a graph G, with the same values of s. Let Q be an advancing path in G that is simultaneously of the forms P1 R1 and R 2 P 2 • Then we can transform Pi into P 2 by repeatedly taking successors. Proof Let P be any subpath of Q of length s, other than P 2 • Clearly P is an s-route. By replacing each edge-term and vertex-term of P by the next edge-

AUTOMORPHISMS

59

term or vertex-term respectively in Q, we obtain a successor of P. Starting with P = Pi and repeating the process sufficiently often, we obtain P 2 • 1.49. Let G be a connected graph having at least one monovalent vertex. Let P be ans-route in G. Then, by repeatedly taking successors, we can transform P into an s-route with a monovalent terminus. Proof We can find a I-route R in G with a monovalent terminus. By 7.46, with Pas Pi and Ras P 2, we can find an advancing path U in G, of the form PQ, that has a monovalent terminus. We can write U = Ui Qi, where Qi is an s-route with a monovalent terminus. The theorem follows, by 7.48.

7.5. s-transitive graphs. A graph G is s-transitive, for a given positive integers, if it has at least one s-route and if for each ordered pair {P, Q} of s-routes of G there exists an automorphism/ of G such that/P = Q. This means that/ maps the ith term of Ponto the ith term of Q, for each relevant value of i. We dismiss the case of a connected s-transitive graph G having a monovalent vertex with the following remarks. Each s-route of G must have a monovalent origin and terminus, by 7.42 and 7.49. Hence G contains no arc whose length exceeds s. Moreover, G contains no polygon, by 7.43, and is therefore a tree. 7.51. A polygon is s-transitive for every positive integers. Proof Let H be any polygon, and s any positive integer. Has at least one s-route, by 7.43. Let (ai, Ai, bi)Qi and (a2 A2, b.;)Q2 bes-routes of H. By 6.45 there is an automorphism/of Hthat maps (ai, Ai, bi) onto (a 2, A 2 , b2) . Then / maps (ai, Ai, bi)Qi onto an s-route (a2, A2, b.;)R of H, and this s-route is identical with (a 2 , A 2, b.;)Q 2 , by 7.43.

7.52. Let G be a connected s-transitive graph that has no monovalent vertex. Then G is k-transitive for each integer k satisfying 1 < k < s. Moreover, G is transitive on its edges and vertices. It is a regular graph of valency > 2. Proof Any k-route, with 1 < k < s, can be extended to form an s-route, by 7.43. It follows that G is k-transitive. Since G is I-transitive, it is transitive on its edges. Since G is connected and not a vertex-graph, each vertex is incident with an edge and therefore the origin of a I-route. Hence the Itransitivity of G implies transitivity on the vertices. G is therefore regular. Since it has at least one edge, but no monovalent vertex, its valency is at least 2. 1.53. Let G be a connected regular graph, of odd valency 2m + 1, which is transitive on both its edges and its vertices. Then G is I-transitive. Proof Let P = (a, A, b) be any I-route. Let M be the class of all I-routes R such that/P = R for some f EA( G). It follows from this definition that each

60

CONNECTIVITY IN GRAPHS

element of A(G) maps Minto M. If R is any I-route of G, then either REM or R- 1 EM, since G is transitive on its edges. Assume that there is no I-route R of G such that both Rand R- 1 are in M. For each x E V(G) let ,\(x) denote the number of I-routes of M having x as origin. For any two vertices x and y of G there is an automorphism g such that gx = y . Both g and g- 1 map M into itself. Hence ,\(x) = ,\(y). We deduce that the function ,\(x) has the same value, ,\ say, for all vertices x of G. It follows that ,\oco(G) = oc1(G). Hence ,\ = m + ½, by 1.21, which is contrary to the definition of,\ as an integer. The above contradiction shows that there is a I-route R of G such that R EM and R- 1 EM. Let T be any member of M. There are automorphisms /i,/2, and g of G such that/1 P = R,/2 P = R- 1 , and gP = T. Hence g/2-111 P = g/2 - i R = gP- 1 = T- 1 EM. Thus M includes all the I-routes of G, Since P can be chosen arbitrarily, the theorem follows. It is not known if Theorem 7.53 extends to connected regular graphs of even valency. The next theorem provides a test for s-transitivity. 7.54. Let G be a connected graph, and let S be ans-route of G whose terminus is not monovalent. For each successor T of S let there be given an automorphism f of G such that JS = T. Then G is s-transitive and has no monovalent vertex. Proof Let M be the class of all s-routes Z of G such that Z = gS for some g E A(G). Suppose some Z EM has a successor Z 1 • Then g- 1 maps Z onto S, and Z 1 onto a successor T of S. But T = JS for some f E A( G). Hence Z 1 = gfS EM. We deduce that M includes all the s-routes that can be derived from S by repeatedly taking successors. Since S has a terminus that is not monovalent, no member of M can have a monovalent terminus. Hence G has no monovalent vertex, by 7.49. Moreover, M includes all the s-routes of G, by 7.46 and 7.48, unless G is a polygon. The theorem follows, by 7.51.

As an example, we apply 7.54 to the Heawoodgraph, shown in Figure 7.5A. Since this graph is simple, we can specify any s-route by the sequence of its vertex-terms. Let S denote the 4-route (1, 2, 3, 4, 5). Then S has just two successors, T1 = (2, 3, 4, 5, 6) and T 2 = (2, 3, 4, 5, 14). It is easily verified that there is an automorphism

Ji =

(1, 2, 3, 4, 5, 6, 11, 10) (14, 7, 12, 9) (13, 8)

mapping S onto Ti, and an automorphism

/2 =

(1, 2, 3, 4, 5, 14) (10, 7, 12, 9, 6, 13) (11, 8)

mapping S onto T2 • Hence the Heawood graph is 4-transitive, by 7.54.

61

AUTOMORPHISMS

,,

4

FJGURE 7.5A.

7.6. Girth. Let G be a graph that is not a forest, that is, which contains a polygon. We define its girth y(G) as the least integer m such that G contains an m-gon. 7.61. Let G be a connected s-transitive graph that has no monovalent vertex and is not a polygon. Then s

< ½,,(G) + 1.

Proof We note that G is not a forest, by 3.35. Hence y(G) is defined. There is a circular path P in G of length y( G). Suppose first thats > y(G). By extending Pin G(P) we can obtain ans-route

P1 in which the last edge-term is a repetition of another, by 7.41 and 7.43. Write P 1 = Qi(x, A , y). We have val(G, x) > 2, by 7.52, since G is not a polygon. Hence there is an edge B of G that is incident with x but not in E(P). Let its other end be z. The path P 2 = Q 1(x, B, z) is an s-route in which the last edge-term is not a repetition. Hence no automorphism of G maps P 1 onto P 2 , contrary to hypothesis. We deduce that in facts < y(G).

62

CONNECTIVITY IN GRAPHS

In the polygon G(P) we can construct a simple path R of length s - l. The terminus x of R is at least trivalent, by 7.52. Hence we can find two distinct s-routes R(x, A, y) and R(x, B, z) of G, where A E E(P) and B ¢ E(P). The origin a of R is joined to x by a simple or circular path RA in G(P) whose length is y(G) - s + land which traverses A. We may suppose RA simple, for ifs = l the theorem is trivially true. Then, bys-transitivity, a is joined to x by a simple path R 8 in G whose length is y(G) - s + land which traverses B. Since the arcs G(RA) and G(R8 ) are distinct, their union contains a polygon having not more than 2y(G) - 2s + 2 edges, by 3.55 and 3.74. Hence y(G)

< 2y(G)

- 2s

+ 2.

The theorem follows. In Section 7.5 we found the Heawood graph to be 4-transitive. Hence, by 7.61, its girth is at least 6. But inspection of Figure 7.5A reveals the hexagon (2, 3, 4, 5, 6, 7). We deduce that the girth is 6. We can now use 7.61 to show that the Heawood graph is not 5-transitive. 7.62. Let G be a connected s-transitive graph having no monovalent vertex. Then ifs < y(G), the s-routes of G are all simple paths. Jf in addition G is not a polygon, then the s-routes are simple paths if either s > 3 or y(G) > 3. Proof Ifs< y(G), we can construct a simple path P, oflength s, in a y(G)gon of G. Then Pis ans-route of G. Hence all the s-routes of Gare simple, by s-transitivity. Suppose G is not a polygon. Ifs > 3, we have y(G) > 2s - 2 > s, by 7.61. If instead y(G) > 3, we have y(G) = ½Y(G) + ½r(G) > (s - 1) + 1 = s, again by 7.61. The theorem follows.

7.7. s-regular graphs. A general problem of great difficulty now arises, that of classifying the possible symmetries of a k-valent connected graph. The cases k = 0 and k = 1 are of course trivial, and the case k = 2 has been disposed of in Section 7.2. Very little is known about the cases k > 4, but some interesting results have been obtained in the case k = 3. In what follows we confine our attention to this last case. Let G be a connected trivalent regular graph. As a regular graph it is nonnull. It therefore has at least one link. Hence if it is transitive on its edges it can have no loops. We say G is s-regular if it is s-transitive and for each ordered pair {P, Q} of s-routes there is just one automorphism/ of G such that/P = Q. 7.71. Suppose G is s-transitive but nots-regular. Then for each s-route S of G there existsf e A(G) such that JS= Sand/interchanges the two successors of S. (See 7.47.)

63

AUTOMORPHISMS

Proof Let S be any s-route of G, with successors T and T'. By hypothesis, there are s-routes Z 1 and Z 2 of G and distinct automorphisms x and y such that Z 1 = xZ2 = yZ2• There exists z E A(G) such that Z 1 = zS. Write g = z- 1xy- 1z. Then gS = S. But g -=I= I, for otherwise we would have

y = zgz- 1y = zz- 1xy- 1zz- 1y = x. We now show that there is ans-route S1 of G such that S1 , but no successor of Si, is invariant under g. For suppose there is no such s-route. Then, since S is invariant under g, one of its successors is invariant under g. Hence both successors of Sare invariant under g. Similarly, all their successors are invariant under g, and so on. Thus every s-route of G is invariant under g, by 7.46 and 7.48. Since we can construct ans-route by extending an arbitrary I-route, by 7.43, it follows that g = I, which is a contradiction. Having established the existence of S 1 we denote its successors by T1 and T{. There exists h E A(G) such that hS = S 1• Then h maps the successors of S onto the successors of S1• We can adjust the notation so that hT = T1 and hT' = T{. By the definition of S1 we have gS1 = Si, gT1 = T{ and gT{ =T1 • Write/= h- 1gh. Then JS= S,fT = T', andfT' = T. 7.72. Suppose G is s-transitive but nots-regular. Then G is (s Proof By extending ans-route we can obtain an (s

+

+

1)-transitive.

1)-route

of G. Let its two successors be Y = (ai, ..., a,, A,+ 1 , a,+ 1 , A,+ 2 , a,+J, Y'

= (ai, . .. , a., A,+i, a,+i,

B,+ 2, b,+J-

Let k be one of the automorphisms of G satisfying Then kA,+ 1 , being incident with a,+ 1 but not A,+ 1 , is either A,+ 2 or B,+2• We can adjust the notation so that kA,+ 1 = A,+2· This implies that Y = kX. By 7.71 there existsf E A(G) such that f A,+2 = B,+ 2, andfB,+ 2 = A,+ 2• This implies that Y' = fY = fkX.

Thus X can be transformed into either of its successors by automorphisms of G. Hence G is (s + 1)-transitive, by 7.54. 7.73. Suppose G is transitive on its edges and its vertices. Then G is s-regular for some positive integer s.

64

CONNECTIVITY IN GRAPHS

Proof Since G is not a forest, its girth y( G) is defined. G is I-transitive, by 7.53. Hence, by 7.61, there is a greatest positive integer s such that G is s-transitive. Then G is s-regular, by 7.72.

We have seen that tlie Heawood graph is 4-transitive but not 5-transitive. Hence it is 4-regular, by 7.72. Examples of s-regular graphs, for all values of s from 2 to 5, are given in the next chapter. It is known that there is no s-transitive trivalent graph with s > 5 (cf. ref. 6). It is difficult to find I-regular trivalent graphs. The first known example is one due to R. Frucht (4), which has 432 vertices and a girth of 12. BIBLIOGRAPHY

1. H. S. M. Coxeter, Self-dual configurations and regular graphs, Bull. Amer. Math. Soc., 56 (1950), 413-455. 2. R. Frucht, Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Comp. Math., 6 (1938), 239- 250. 3. --Graphs of degree three with a given abstract group, Can. J. Math., 1 (1949),

365-378. 4. - - A one-regular graph of degree three, Can. J. Math., 4 (1952), 240-247. 5. I. N. Kagno, Linear graphs of degree , 6 and their groups, Amer. J. Math., 68 (1946), 505-520. Corrections, Amer. J. Math., 69 (1947), 872. 6. W. T. Tutte, A family of cubical graphs, Proc. Cambridge Phil. Soc., 43 (1947). 459-474. 7. - - - On the symmetry of cubic graphs, Can. J. Math., 11 (1959), 621--

Given a non-null subgraph Hof G we define R/H), where j is any nonnegative integer, as the set of all vertices y of G such that d(H, y) = j. We also write S/H) for the set of ally E V(G) such that d(H, y) ,< j. Thus Ro(H) = So(H) = V(H). 8.14. If x and y are adjacent vertices of G, and x where lj - k I < 1 (by 8.13).

E

R/H), then y

E

Rk(H),

We define T/H), for each positive integer j, as the subgraph of G whose vertices are the members of S/H) and whose edges are the edges of G incident with members of Sj_i(H). Theorem 8.14 ensures that the condition of 1.32 is satisfied. We extend this definition by writing To(H) = H . 65

66

CONNECTIVITY IN GRAPHS

8.15. Let 'Ai, j > 0, be the number of edges of G, not in E(H), that have both ends in Rj(H), and let /Li be the number ofedges ofG that have one end in Rj(H) and the other in Ri+i(H). Then

L

JCEl land simple if y(G) > 2. 8.31. Let H be a connected non-null subgraph of G. Let j be a positive integer such that 2j + d(H) < y(G). Suppose that y E R/H). Then val(T/H), y) = l. This theorem follows from 8.161 to 8.171. In the next few theorems we use the notation of 8.15. 8.32. Let H be a connected non-null subgraph of G. Let j be an integer such that j > 0 and 2j + d(H) + l < y(G). Then 'Ai= 0 (by 8.172).

69

GIRTH

8.33. Let H be a connected non-null subgraph of G. Let j be a positive integer such that 2j + d(H) < y(G). Then the following propositions hold:

=

(i) /LJ-1

I Rj(H) I •

j-1

(ii)

(iii)

L /L1 = a.i(Tj(H)) -

a. 1(H).

i=O

J-1

L /Li

i=O

= I Sj(H) I - a.o(H)

Proof. Formula (i) follows from 8.15 and 8.31. It yields formula (iii) on summation. Formula (ii) is a consequence of 8.14 and 8.32.

8.34. Let H be a non-null subgraph of G that is a tree. Let j be a non-negative integer such that 2j + d(H) < y(G). Then Tj(H) is a tree. Proof. By 8.31 every non-null detached subgraph of Ti(H) has a vertex in common with H. Hence the only component of Till) is the one containing H. Thus po(Tj(H)) = 1. We now have Pi(T1(H))

= rx1(Tj(H)) - I Sj(H) I + 1, = a.i(H) - a.o(H) + 1, = Pi(H) = 0,

by 3.21, by 8.33, by 3.23.

Hence Tj(H) is a tree, by 3.32. 8.35. If G is a connected graph that is not a tree, then y(G)

< 2d(G).

Proof. Choose vertices x and y of G such that d(x, y) = d(G) . Apply 8.34 with H = [x] andj = d(G). Since G is not a tree, we must have 2d(G)

> y(G) -

d([x])

=

y(G).

8.36. Let G be a regular k-valent graph, where k > 1. Let H be a connected nonnull subgraph of G. Then the following propositions hold: (i) ka.o(H) provided that 2

+ d(H)
k(k - 2)- 1 {(k - l)' - 1} + 1.

But

if y( G) is an even number 2t, then ao(G)

> 2(k -2)- 1 {(k

- 1)' - 1}.

Proof. The theorem is trivially true fort = 0. For other values oft we obtain the first inequality by putting j = t in 8.37(ii), and the second by putting ; = t -1 in 8.38(ii). 8.4. Graphs of girth < 4. We now discuss the problem of finding a regular graph G, of given girth y(G) and valency k > 3, that has the least possible number of vertices. The case y(G) = 1 is trivial. If k is an even number 2t, then G is made up of one vertex and t loops. If k is an odd number 2t + 1, then G has just two vertices. They are joined by an odd number l < k oflinks, and each is incident with exactly ½(k - /) loops. The case y( G) = 2 is also trivial. G has just two vertices, and they are joined by exactly k links. We deal with the cases y( G) = 3 and y(G) = 4 in the following two theorems. 8.41. A regular graph of girth 3 and valency k > 3 has at least k + 1 vertices. The only such graph with exactly k + 1 vertices is the (k + 1)-clique.

Proof. The necessity for k + 1 vertices follows from 8.39. A simple graph of k + l vertices can be k-valent only if each vertex is joined to all the others, that is, only if it is a (k + 1)-clique. On the other hand, the (k + 1)-clique is clearly k-valent and of girth 3. 8.42. A regular graph of girth 4 and valency k > 3 has at least 2k vertices. The only such graph with exactly 2k vertices is the bipartite graph Kk.k.

71

GIRTH

Proof. Let G be a regular graph of girth 4 and valency k

> 2k, by 8.39.

> 3. Then a 0(G)

Assume that ao(G) = 2k. Choosing a vertex x, we define X as the set of all vertices adjacent to x, and Y as the set V(G) - X. Then I XI = I YI = I Ri(x) I = k, by 8.37. Since G has no loop and no triangle, no edge of G can have both ends in y. Hence, since G is k-valent, each vertex of Y is adjacent to each vertex of X, and no edge has both ends in x. Thus G is identical with K" "' to within an isomorphism. Conversely, let {X, Y} be the bipartition of Kk,t• We can find two distinct vertices x 1 and x 2 of X, and two distinct vertices Yi and y 2 of Y. Since K1 ,1 contains the quadrilateral x 1 Yi x 2 y 2, its girth is at most 4. But K,.," is simple, and it has no triangle, by 8.25. Hence y(K",J = 4. This completes the proof of the theorem. From the foregoing discussion we see that there is a unique m-cage for each of the values 2, 3, and 4 of m. The 2-cage consists of two vertices joined by three links, the 3-cage is the 4-clique, and the 4-cage is the graph K 3 , 3 • The 4-cage is also known as the Thomsen graph. It is shown in Figure 8.4A, where the bipartition is indicated by the distinction between even-numbered and odd-numbered vertices.

6

1.

FIGURE

8.4A.

The 3-route(t ; 2, 3, 4) of Figure 8.4Ais mapped ontg itssuccessors(2, 3, 4, 5) and (2, 3, 4, I) by the automorphisms (l, 2, 3, 4, 5, 6) and (1, 2, 3, 4) (5, 6) respectively. Hence the Thomsen graph is 3-regular, by 7.54, 7.61, and 7.73. The order of the automorphism group is thus equal to the number of 3-routes, which is 6.3.2.2 = 72, since we may choose the origin of a 3-route in ao(G) = 6 ways, the first edge in 3 ways, and each successive edge thereafter in 2 ways only.

72

CONNECTIVITY IN GRAPHS

It is readily verified that the 2-cage and 3-cage are 2-regular. Their automorphism groups have orders 12 and 24 respectively. The graphs defined by the edges and vertices of the cube and the regular dodecahedron are also 2-regular. 8.5. Cages. The problem of finding a graph with given girth y(G) > 5, valency k > 3, and having the least possible number of edges has been solved only fork = 3, and then only for the values 5, 6, 7, and 8 of y(G). In these cases the graph required is an m-cage, as defined in §8.1, where m = y( G). * Let G be an m-cage, where y(G) = m > 5. If mis an odd number 2t + I, we choose a vertex x of G, and if m is an even number 2t + 2, we choose an edge A. We note that, for O l, by 8.31. Let a be any divalent vertex of'½, where 2 l, we have

I Ri+1(x) I < (k -

I

1) Rix)

I,

for, by 8.161, (k - 1) Rix) is not less than the expression on the left of 8.15, and the expression on the right of 8.15 is not less than I Ri+ 1(x) j. The corresponding result in the case j = 0 is

I Ri(x) I < w I R (x) I = 0

We thus have

r

I S,(x) I = L

j-o

8.82. Let m

Rix)

= 1+

> 3 and k > 3 be integers. N

= (k - 1){,\(k, m

r

m - l and d(y, z) > m. Since d(x, z) > q(G), we have val(G, z) = k > 3. Hence there is an edge A of G that is incident with z and is not traversed by every m-gon of G. Let u be the other end of A. We have d(y, u) > d(y, z) - d(z, u) = m - l > q(G), by 4.43. Hence val(G, u) = k. We join x and z by a new edge B, and delete the edge A. The resulting graph, G1 say, is clearly a member of S with the same number of edges as G. Now D(GJ contains all the members of D(G), except possibly x, and in addition it includes u. It follows from the definition of G that I D(G1) I = I D(G) I and x ¢ D(GJ. This implies that val (G, x) = k - l. We can now show that xis not the only member of D(G); for, if it were, the number of vertices of G of odd valency would be odd, contrary to 1.22, since N is even when k is odd. It follows that x and y are distinct. Now y and u are distinct vertices of D(G1). Let P be any simple path from y to u in G1 (if such exists). If Pis a path in G, we have s(P) > d(y, u) > q(G). In the remaining case P traverses B. It then contains, as a subpath, a simple path Q from y to x in G. Then s(P) > s(Q) > q(G). We deduce that, in all c.ases, q(GJ > q(G). But this is contrary to the choice of G. We conclude that in fact D(G) is null, that is, G is a regular k-valent graph. This completes the proof of the theorem. BIBLIOGRAPHY

1. H. S. M. Coxeter, Self-dual configurations and regular graphs, Bull. Amer. Math. Soc., 56 (1950), 413--455. 2. P. Erdos and H. Sachs, Reguliire Graphen gegebener Taillenweite mil minima/er Knotenzahl, Wiss, Z. Univ. Halle, Math.-Nat., 12, no. 3 (1963), 251-258. 3. A. J. Hoffman and R. R. Singleton, On Moore graphs with diameters two and three, I.B.M. J. Res. Develop., 4 (1960), 497-504. 4. W. F. McGee, A minimal cubic graph of girth seven, Can. Math. Bull., 3 (1960), 149-152. 5. W. T. Tutte, A family of cubical graphs, Proc. Cambridge Phil. Soc., 43 (1947), 459--474. 6. - - A non-Hamiltonian graph, Can. Math. Bull., 3 (1960), 1-5. 7. - - Symmetrical graphs and coloring problems, Scripta. Math., 25 (1961), 305-316. 8. - - The thickness of a graph, Nederl. Akad. Wetensch. Proc. 66 (1963), 567577.

9 SEPARATION 9.1. Separable and non-separable graphs. A graph G is separable if it has a vertex x such that G is not [x]-connected. 9.11. Every null graph, vertex-graph, link-graph, and loop-graph is nonseparable (by 2.33 and 2.34). 9.12. Every non-separable graph is connected (by 2.47). We need the following theorem about J-connection. 9.13. Let J be a subgraph of G, and let C be a J-component of G. Then C is a component mod J fi C of G. Proof C is a subgraph of G that is detached mod J n C and is not a subgraph of J n C. lt is also connected mod J n C, by 2.47. The theorem follows.

9.14. Let G be a non-separable graph. Then if G has a loop, it is a loop-graph, and if G has an isthmus, it is a link-graph. Proof Let A be a loop or isthmus of G. Assume that G · {A} is not identical with G. Then there exists a component C of G mod G · {A}, by 2.45. Suppose Chas two distinct vertices of attachment x and y, Then A must be an isthmus with ends x and y . But x and y are joined by an arc L in GA, by 3.62. Since L is connected, they are vertices of the same component of GA, and we have a contradiction. Suppose next that Chas just one vertex of attachment, x say. Then C is an [x]-component of G, by 2.42 and 9.13. Hence G is separable, contrary to hypothesis. In the remaining case C is detached, and G is not connected, contrary to 9 .12.

9.141. If a non-separable graph G has a monovalent vertex x, then G is a link-

graph.

Proof The edge incident with x is an isthmus, having [x] as an end-graph.

84

SEPARATION

85

9.15. Let Hand K be non-separable subgraphs of a graph G, having either a common edge or two common vertices. Then H u K is non-separable. Proof. Suppose that x E V(H u K). The subgraphs H and K are [x]connected, by 2.47 and the definition of a separable graph. Hence H u K is [x]-connected, by 2.32. The theorem follows.

9.16. Let H be a non-separable subgraph of graph G, and let L be an arc in G whose two ends x and y are in V(H) . Then H u L is non-separable. Proof. Suppose that z E V(H u L). We assume first that z is not an internal vertex of L. Then Hand Lare [z]-connected, by 3.61, and therefore Hu Lis [z]-connected, by 2.32. If z is an internal vertex of L, we proceed as follows. L is the union of an arc L 1 joining x and z and an arc L 2 joining y and z, by 3.53. Both L 1 and L 2 are [z]-connected, by 3.61. Moreover, H is [z]-connected, by 2.47. Hence Hu L = Hu L 1 u L 2 is [z]-connected, by two applications of 2.32. The theorem follows.

9.17. Every polygon is non-separable. Proof. Let Kbe a polygon, and let A be an edge of K. We may suppose that

Kis not a loop-graph, by 9.11. Hence Kis the union of the link-graph G · {A}

and its complementary arc, L say. It is. thus non-separable, by 9.11 and 9.16.

The next theorem is due to H. Whitney (2). It is used by him as a tool in inductive proofs.

9.18. Let G be a non-separable graph. Let K be a non-separable proper subgraph

of G having at least one edge. Then we can write G as H u L , where H is a non-separable proper subgraph of G containing K, and L is an arc in G that avoids Hand has both its ends in V(H). Proof. Let Q be the class of all non-separable proper subgraphs of G containing K. Then KE Q. Hence Q has a maximal member H, by 2.11. There exists an H-component C of G, by 2.45. Chas two distinct vertices of attachment, x and y say. For otherwise C f"'I His either null or a vertex-graph, by 2.31. Then G is separable, by 9.12 and 9.13, which is contrary to hypothesis. There is an arc L in C that joins x and y and avoids H. The graph H u L is non-separable, by 9.16, and it contains K. But it is not a member of Q, by the maximality of H . Hence it is not a proper subgraph of G, that is H u L = G.

9.2. Classification of non-separable graphs. Suppose we have a list Ln of all non-separable graphs of up to n > 1 edges. For each G E Ln we may list the graphs of n + 1 edges that can be obtained from G by adjoining an arc that avoids G and has both its ends in V(G). The graphs of the resulting list are all

G

86

CONNECTIVITY IN GRAPHS

non-separable, by 9.16. Apart from isomorphisms they include all the nonseparable graphs of n + l edges, by 9.18. In this application of 9.18 we can take K to be a link-graph. Starting with the loop-graph and link-graph as the only non-separable graphs with a single edge, we can construct, in succession, lists of the nonseparable graphs of 2 edges, 3 edges, 4 edges, and so on. Some of these lists are shown in Figure 9.2A.

•

•

FIGURE

0

9.2A.

9.3. Cyclic elements. A cyclic element of a graph G is a maximal non-null non-separable subgraph of G. Thus a non-null non-separable graph is its own unique cyclic element. Cyclic elements are often called "blocks." In the

87

SEPARATION

pioneering papers of H. Whitney they are called "components." (See, for example, references 2 and 3.) 9.31. The cyclic elements of a non-null graph Gare the vertex-graphs corresponding to its isolated vertices, together with certain non-null reductions of G. Their union is G. Proof Let x be an isolated vertex of G. Then [x] is non-null and non-separable, by 9 .11. But if [x] c K s;;; G, then [x] is detached in K, and K is separable, by 9.12. It follows that [x] is a cyclic element of G. Let H be any cyclic element of G that is not the vertex-graph corresponding to some isolated vertex of G. Suppose Hhas an isolated vertex y. Then H = [yJ, for otherwise H would be disconnected and separable. By the definition of H there is an edge A of G incident with y, and we have H = [yJ c G · {A}. Since G · {A} is non-separable, by 9.11, this contradicts the definition of H as a cyclic element. We conclude that H has no isolated vertex; it is a reduction of Gas defined in 1.3. Every vertex-graph, link-graph, and loop-graph in G is contained in some cyclic element of G, by 9.11. Hence G is the union of its cyclic elements. 9.32. Let A be an edge of a graph G. Then G · {A} is a cyclic element of G and only if A is a loop or isthmus of G.

if

Proof Suppose that A is a loop or isthmus of G. If G · {A} c Ks;;; G, then A is a loop or isthmus of K, by 3.11. Hence K is separable, by 9.14. It follows, by 9.11, that G · {A} is a cyclic element of G. Co~versely, suppose that G · {A} is a cyclic element of G. If A is not a loop or isthmus of G, then G · {A} is a proper subgraph of some polygon K of G, by 3.74. This is contrary to the definition of G · {A} as a cyclic element of G, by 9.17. 9.33. If Hand Kare distinct cyclic elements of G, then H vertex-graph.

n

K is either null or a

Proof If the theorem fails, Hu K is non-separable, by 9.15. But then H = H u K = K, by the definition of a cyclic element. As a corollary we have 9.331. If A is an edge of G, it is an edge ofjust one cyclic element of G (by 9.31 and 9.33). We describe a subgraph Hof Gas cyclically complete if it is a union of cyclic elements of G. The null subgragh of G is counted as cyclically complete, being regarded as a null union of cyclic elements. G is a cyclically complete subgraph of itself, by 9.31.

88

CONNECTIVITY IN GRAPHS

9.34. Let Q be a class of cyclic elements of G, and let Ube the union of its members. Let H be any non-null non-separable subgraph of U. Then there exists KE Q such that H s;; K. Proof Either His a vertex-graph or it has an edge A, by 9.12. In the first case the theorem is trivial. In the second there exists KE Q such that A E E(K). But then H u K is non-separable, by 9 .15, and so H s;; K by the definition of a cyclic element. 9.35. Let H be a cyclically complete subgraph of G. Then H can be expressed in just one way as a union of cyclic elements of G. Proof Suppose Q and Q' are two classes of cyclic elements of G, each having Has the union of its members. If KE Q, we have KE Q', and conversely, by 9.34. Hence Q = Q'. 9.36. Let H be a cyclically complete subgraph ofG. Then the cyclic elements of H are those cyclic elements of G that are contained in H. Proof Let Q be the class of all cyclic elements of G that are contained in H . Each of these is clearly a maximal non-separable subgraph of Hand is therefore a cyclic element of H . But His the union of the members of Q. It can have no other cyclic elements, by 9.35. 9.37. Let J be a subgraph of G that is either null or a vertex-graph. Let C be a ]-component ofG. Then C is cyclically complete in G. Proof Let Q be the class of all cyclic elements K of G such that K (i C is not a subgraph of J. Let Ube the union of the members of Q. If KE Q, then the graph K (i C has at most one vertex of attachment in K, and each such vertex is in V(J), by 1.431 and 1.432. Since K is non-separable, we deduce that K s;; C. Hence U s;; C. The complement of U in C is a subgraph of J, by 9.31. It is therefore null, by 2.42. Hence C is the cyclically complete subgraph U of G. 9.38. Let H be a non-separable subgraph ofG having at least one edge. Then H is a cyclic element of G if and only if no H-component of G has more than one vertex of attachment. Proof Suppose some H-component C of G has two distinct vertices, x and y, of attachment. Then there is an arc Lin C that joins x and y and avoids H, by 3.62. But now Hu Lis non-separable, by 9.16, and so His not a cyclic element of G. Conversely, suppose His not a cyclic element of G. Then there is a subgraph H' of G that is minimal with respect to the property of being non-separable and having Has a proper subgraph. By 9.18 we can write H' =Hu L, where L

SEPARATION

89

is an arc in G that avoids Hand has its ends x and yin V(H). But Lis contained in an H-component C of G, by 3.61, and Chas at least the two vertices of attachment x and y. Let A be an edge ofa graph G. We can use the following algorithm to determine the cyclic element K of G such that A E E(K). (See 9.331.) We first find a non-separable subgraph H of G having A as an edge. Initially this may be G · {A}. For each vertex x of Hwe construct the recessional sequence R(H; x) as defined in 4.4. For some x we may find a vertex y of H that is distinct from x but belongs to a member of R(H; x). We can then find an arc Lin G that joins x and y and avoids H, by 4.44. We replace H by the non-separable graph H u L and repeat the process. When the process terminates, H is replaced by a non-separable subgraph K of G having A as an edge and such that no K-component of G has more than one vertex of attachment, by 4.47. This is the required cyclic element, by 9.38. 9.4. Cut-vertices. Let x be a vertex of a graph G. We write {3(G, x) for the number of cyclic elements of G having x as a vertex. This number is positive, by 9.31. Ifit exceeds 1, we say that xis a cut-vertex of G. In this case [x] is not a cyclic element of G.

9.41. Let K be any cyclic element of G. Then a vertex x of K is a cut-vertex of G if and only if it is a vertex of attachment of K. Proof. Suppose that x E W(G, K). Then xis incident with an edge A of G not in E(K). But A is an edge of a cyclic element of G other than K, by 9.31. Hence xis a cut-vertex of G. Conversely, suppose xis a cut-vertex of G. Then it is a vertex of two distinct cyclic elements J and K of G. Since J is not a subgraph of K, it is not a vertexgraph. Hence, by 9.31, it has an edge A incident with x. But A ¢ E(K), by 9.331. Hence x E W(G, K).

9.42. Let x be a vertex of a graph G, and let C be an [x]-component of G such that x E V(C). Then C containsjustonecyclicelement KofGsuch that x E V(K). Proof. C contains one such cyclic element K of G, by 9.37. Suppose J is another. Since xis not an isolated vertex of G, by 2.42, neither J nor K is a subgraph of[x], by 9.31. ButJ n K = [x], by 9.33. Since Cis [x]-connected, each of J and K must have a vertex distinct from x. By 3.65 there is an arc Lin C, with ends a and b, such that L n J = [a], L n K = [bJ, and L n [x] = n. There is an arc Min J joining a and x, by 9.12 and 3.62. Then L u Mis an arc in C that joins b and x and avoids K, by 3.52. The graph K u L u Mis nonseparable, by 9.16, and has Kasa proper subgraph. This is contrary to the definition of K as a cyclic element of G.

90

CONNECTIVITY IN GRAPHS

9.5. Cut-forests. Let G be any graph. We define a second graph Fas follows.

V(F) is the union of two disjoint sets v,, and Vw of "black" and "white" vertices respectively. v,, is in 1-1 correspondence with the set of cyclic elements of G, and Vw with the set of cut-vertices of G. If Xis a cyclic element or cutvertex of Gandy is the corresponding member of V,, or Vw respectively, we writey = /Xand X = J- 1y. Fisasimplegraphhaving {V,,, Vw}asa bipartition. Two vertices x e v,, and y e Vw are joined by an edge of F if and only ifJ- 1y e V(J-lx). . The construction of Fis illustrated in Figure 9.5A, in which corresponding objects are denoted by the same letter.

Cs

0

G

FIGURE

Cs

a,

• c6

F

9.SA.

9.51. Each white vertex of Fis at least divalent (by the definition of a cutvertex). We say that a subgraph F 1 of Fis white-attached if it has no isolated white vertex and no black vertex of attachment. Thus the null subgraph of F is white-attached. Moreover, F itself is white-attached in F, by 9.51. We write P for the class of white-attached subgraphs of F, and Q for the class of cyclically complete subgraphs of G. Suppose that G1 e Q. Let /G1 denote the subgraph of F generated by the black vertices corresponding to those cyclic elements of G that are contained in Gi, and the white vertices corresponding to those cut-vertices of G that belong to V( G1). Then if V(/G1) includes a black vertex x, it includes also every white vertex adjacent to x. Moreover, every white vertex of/G1 is adjacent to some black vertex of/G1 • Thus/G1 is white-attached.

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91

9.52. J/G1 e Q, then/G1 eP. We thus have a mapping¢,, G1 -+ /Gi, of Q into P. 9.53. ¢, is a 1-1 mapping of Q onto P. Proof. Suppose G1 and G2 are members of Q such that /G1 = JG,.. Then Gi and G2 contain the same cyclic elements of G, and are therefore identical. Hence¢, is a 1-1 mapping of Q onto a subset P' of P. Now let F 1 be any member ofP. Let Gibe the union of those cyclic elements of G that correspond to the black vertices of F 1• Then G1 e Q and ¢,G1 = /G1 e P, by 9.52. But/G1 has the same black vertices as F 1, by 9.35. Hence F 1 = /G1• The theorem follows.

9.54. Suppose that G1 e Q. Let x be any vertex of attachment of G1 in G. Then x is a cut-vertex of G, and fx is a vertex of attachment of/G1 in F. Proof. xis a vertex of some cyclic element J of G contained in G1 • It is also incident with an edge of G not in E(GJ. It is thus a vertex of a cyclic element K of G that is not a subgraph of Gi, by 9.31. Accordingly xis a cut-vertex of

G.

The vertex/x of Fis adjacent to fK ¢ V(fGJ. Hence it is a vertex of attachment of/Gi in F.

9.55. Suppose that Fi e P. Let q be a vertex of attachment ofF1 in F. Then J- 1q isa vertex ofattachment oJ,f,- 1Fi in G.

Proof. q is necessarily white. It is joined to a black vertex b ¢ V(Fi). Accordingly, the vertex 1-iq of ,f,- 1Fi is a vertex of the cyclic element 1-ib of G, andJ-ib is not a subgraph of ,f,-iFi. HenceJ- 1q is not an isolated vertex of J- 1b, by 9.31. But no edge ofJ- 1b is an edge of the cyclically complete subgraph ,f,- 1 F 1 of G, by 9.331. ThusJ- 1q is a vertex of attachment of ,f,- 1Fi.

It is clear that ¢, and its inverse preserve inclusion relations among subgraphs. By 9.54 and 9.55 they also preserve the property of detachment. Since the components of Fare white-attached and those of Gare cyclically complete, by 9.37, it follows that¢, induces a 1-1 mapping of the set of components of G onto the set of components of F. We thus have 9.56. p 0(G)

= Po(F).

9.57. Fis a forest. Proof. Let A be any edge of F, with white end w and black end b. Let G1 be the [J- 1 w]-component of G that contains the non-separable, and therefore [J-iw]-connected, subgraphJ-ib. Write Fi = /G1 • (See 9.37.) By 9.55 Fi has no vertex of attachment in F other than w. Moreover, A is the only edge of Fi

92

CONNECTIVITY IN GRAPHS

incident with w, by 9.42. On deleting from Fi the edge A and the vertex w we obtain a subgraph F2 of Fi that is detached in F:,.. It follows that A is an isthmus of F. Since A can be chosen arbitrarily, Fmust be a forest. We call Fa cut-forest of G. We conclude this section by using the notion ofa cut-forest in the proofs of two general theorems. We denote the number of cyclic elements of a graph G by v( G). We describe a cyclic element as terminal if it had just one vertex of attachment. 9.58.

lf G is connected and separable, it has at least two terminal cyclic elements.

Proof. G has at least two cyclic elements, by 9.31. Any cut-forest F of G is a tree with at least two vertices, by 9.56 and 9.57. Hence F has at least two monovalent vertices, by 3.35. These correspond to terminal cyclic elements of G, by 9.41 and 9.51. 9.59 (Harary, ref. 1). v(G)

= L

xeV(G)

{/J(G, x) - l}

+ Po(G).

Proof. {3(G, x) - 1 is non-zero only if x is a cut-vertex of G. The above equation may therefore be written, in terms of a cut-forest F of G, as follows:

I Vb I = xeVw L {val(F, x)} - I Vw I + Po(F). (See 9.56.) But this equation is equivalent to

ao(F)

= «i(F) + Po(F),

which is true by 3.23.

9.6. A contractive mapping. Let A be an edge of a graph G. We can derive another graph G~ from Gin the following way. We introduce a single new vertex z, and denote the ends of A by x and y. For each v e V(G) we write fAv = v if vis not x or y, and/Av = z if vis x or y. We also write/,4.B = B for each edge B of G other than A, and put fAA = z. The definition of GA is as follows: (i) V(G~) = (V(G) - {x, y}) v {z}; (ii) E(GA)

=

E(G) - {A};

(iii) if Be E(GA) and the ends of Bin Gare u and v, then its ends in G~ are/,4.u and/,4.v. The construction of G~ is illustrated in Figure 9.6A.

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SEPARATION

G; FIGURE

9.6A.

We can regard/,. as the symbol of a mapping of G onto G~, by 6.1. The mapping is contractive, by 6.3. It is often possible to prove inductively a theorem about a general graph G by making use of the properties of o_:. and G~. Usually we depend on the following auxiliary theorems. 9.61. Let H be a subgraph of o_:. such that

I W(G,:., H) I < I W(G, H) I , I W(G,:., H) I< I W(G, Hu (G · {A})) j. Then A has one end x in V(H) and one end y not in V(H). Moreover, H) and

xi W(G_:.,

W(G, H)

=

W(G_:., H) u {x}.

Proof. By 1.43 we have

W(G, H)

= =

W(G_:., H) u {V(H) n W(G, G_:.)} W(G_:., H) u Y,

where Y is the set of all ends of A in V(H). We also have W(G, Hu (G · {A}))

= W(G_:., Hu L) u

{V(H u L)

n

W(G, G)},

by 1.43, where Lis the edgeless graph whose vertices are the ends of A,

= W(G_:., Hu L) = W(G.'.t, Hu M), where M is the edgeless complement of H in H

u L,

£ W(GA, H) u Z, where Z is the set of all ends of A not in V(H), by 1.44. By hypothesis, both Y and Z must be non-n~ll. The theorem follows.

94

CONNECTIVITY IN GRAPHS

9.62. Let J be a subgraph of G~ such that

I W(G~, J) I < I W(G,J,.- 1J) 1. Then the ends x and y of A are distinct. Moreover, z ment of J in G~, and W(G,f,.-lJ)

=

= f,.A is a vertex ofattach-

(W(G~,J) - {z}) u {x,y}.

This theorem is a consequence of 6.25. 9.63. Let A be an edge of a graph G. Let H be a subgraph of G;_ satisfying the conditions of 9.61. Let J be a subgraph of G~ satisfying the conditions of 9.62. Write K = f,. - 1J. Then G has an edge not in E(H u K). Moreover, min{ I W(G, Hu K)

I , I W(G, H n K) I } < max{ I W(GA, H) I , I W(G~, J) I }.

Proof. By 9.61 A has one end x in V(H) and one end y not in V(H). Now y is a vertex of attachment of Kin G, by 9.62. Hence there is an edge B of G, not in E(K), that is incident with y. Clearly B ¢ E(H). Hence B ¢ E(H u K), Since x ¢ W(G;., H), by 9.61, all the edges of G incident with x, except A, belong to E(H). But A E E(K). Hence xis not a vertex of attachment of H u K. Thus, in the notation of 1.46, q(H, K) > 1. We have

I W(G, Hu

K) I + I W(G, H () K) I < I W(G, H) I + I W(G, K) I - 1,

= I W(GA, H) I + I W(G~, J) I + I,

by 1.46,

by 9.61 and 9.62. The theorem follows. We now apply the foregoing results to the theory of non-separable graphs. We require the following lemma. 9.64. Let G be a non-separable graph. Let K be a subgraph ofG such that E(K) and E(G) - E(K) are both non-null. Then I W(G, K) I > 2. Proof. If W(G, K) is null, then G is disconnected, contrary to 9.12. If

W( G, K) consists of a single vertex x, then G is not [x]-connected, and therefore

G is separable.

9.65. Let A be an edge of a non-separable graph G. Then either G;_ or G,; is non-separable. Proof. Suppose G has only the one edge A. Then G is a link-graph or a loop-graph, by 9.12. Hence G~ is a vertex-graph, and so non-separable. In the remaining case G has an edge C distinct from A .

95

SEPARATION

Assume that G.:._ and G~ are both separable. G.:._ is connected, by 9.14. It has a vertex u such that G.:._ is not [u]-connected, Let H be the [u]-component of G.:._ having C as an edge. Since H is not a component of G.:._, it has u as a vertex of attachment. Hence G.:._ has an edge D1 not in E(H). We note that I W(G.:._, H) I = I. But I W(G, H) I > 2 and I W(G, Hu (G · {A})) I > 2, by 9.64. Thus H satisfies the conditions of 9.61. G~ is connected, by 6~ 13. It has a vertex w such that G~ is not [w]-connected. Let J be the [w]-component of G; having C as an edge. Write K = f,1.- 1J. Since J is not a component of G~, it has was a vertex of attachment, and G~ has an edge D 2 not in E(J). We note that I W(G~, J) I = I. But I W(G, K) I > 2, by 9.64. Thus J satisfies the conditions of 9.62. We now have min{ I W(G, H u K) I , I W(G, H by 9.63. But C is an edge of H by 9.63. Hence min{ I W(G, H

f"\

u

f"\

K)

I } < 1,

K, and there is an edge of G not in E(H K)

I , I W(G, H

f"\

u K),

K) I } > 2,

by 9.64. This contradiction establishes the theorem. 9.7. Cyclic connection. Two distinct edges A and B of a graph Gare said to be cyclically connected in G if there is a polygon K of G traversing both A and B.

9.11. Let x and y be distinct vertices, and A an edge, of a non-separable graph G. Then there is an arc L of G that joins x and y and traverses A. Proof. If a:1{G) < 1, the theorem is trivially true. Assume as an inductive hypothesis that it is true for a:1( G) < q, where q is an integer > 1. Consider the case a:1(G) = q. By 9.18 we can write G = H u J, where A E E(H), His non-separable, J is an arc that avoids H but has its ends a and b in V(H), and a =j; y. If x and y are both in V(H), the theorem holds, since it is true for H by the inductive hypothesis. If just one of x and y is an internal vertex of J, then we can suppose it to be x. By the inductive hypothesis there is an arc M in H that joins a and y and traverses A. There is an arcNinJthatjoinsa and x and does not pass through b, by 3.53. Then M u N is an arc L having the required properties, by 3.52. In the remaining case, x and y are both internal vertices ofJ. By the inductive hypothesis a and bare joined in H by an arc M traversing A. The graph M u J is a polygon K, by 3. 72. One of the residual arcs of x and y in K traverses A and may be taken as L. The theorem now follows in general by induction.

96

CONNECTIVITY IN GRAPHS

9.72. Two distinct edges A and B of a graph Gare cyclically connected in G if and only if they belong to the same cyclic element of G. Proof. The necessity of this condition follows from 9.17. To prove sufficiency suppose A and B belong to the same cyclic element Hof G. Let the ends of B be a and b. There is an arc Lin H that joins a and band traverses A, by 9. 71. L does not traverse B, by 3.64. Adjoining B to L, we obtain a polygon in H that traverses both A and B, by 3.72. This theorem is due to H. Whitney (2). Suppose we wish to find a polygon traversing two given distinct edges A and Bin a non-separable graph G. Suppose there is a third edge C. It is not difficult to show that a solution for G can be derived from any solution for Ge or G~, whichever is non-separable. (See 9.65.) We can determine whether or not Ge is separable by using the algorithm described at the end of §9.3. Repeating this procedure sufficiently often, we arrive at a graph for which the solution is evident. BIBLIOGRAPHY 1. F. Harary, An elementary theorem on graphs, Arner. Math. Monthly, 66 (1959),

405--407.

2. H. Whitney, Non-separable andplanar graphs, Trans. Arner. Math. Soc., 34 (1932), 339-362. . 3. - - Congruent graphs and the connectivity ofgraphs, Amer. J. Math., 54 (1932), 150--168.

10 THREE-CONNECTION 10.1. Connectivity. Let k be a non-negative integer. We say that a graph G is k-separated if it has a non-null proper subgraph H with the following properties :

(i) min{ai(H), ai(H)} (ii)

I W(G, H) I = k.

> k,

We denote the class of all such subgraphs H by QtCG). The connectivity ,\(G) of G is the least integer k, if such exists, such that G is k-separated, that is, such that Q,.{G) is non-null. If there is no such integer k, we write ,\(G) = co. We say that G is n-connected, where n is any positive integer, if n < ,\(G). Other writers give a slightly different meaning to the term "n-connected." 10.Il. If G is n-connected and HE QnCG), then HE QnCG), W(G, H) W(G, H), and R

= H.

=

Proof W(G, H) s W(G, H), by 1.51. Hence HE Q,.(G) for some k< n. Since G is n-connected, we have k = n. Hence W(G, H) = W(G, H) . It follows, by the definition of the complement of a subgraph, that R = H.

if and only if it if it is non-separable.

I0.I2. G is I-connected

only

is connected, and 2-connected if and

Proof By the above definitions G is 0-separated if and only if it has a detached non-null proper subgraph H . Hence G is I-connected if and only if it is connected. Now suppose G is connected. If it is I-separated, choose HE Qi( G). Then if W(G, H) = {x} the graph G is not [x]-connected. Thus G is separable. Conversely, assume that G is separable. It has at least two [x]-components for some vertex x. Each of these [x]-components, being non-detached in G, has an edge incident with x. Hence each of them belongs to Qi( G). Accordingly, G is I-separated. We deduce that G is 2-connected if and only if it is nonseparable.

97

98

CONNECTIVITY IN GRAPHS

In view of this result we may regard the theory of k-connected graphs as a generalization of the theories of connected and non-separable graphs. 10.13. IfG is n-connected and His a subgraph ofG such that I W(G, H) I = n and min{cx1(H), cxi(H)} > n - l, then His connected.

Proof. If H is not connected, it has complementary detached non-null subgraphs Kand L . We have

I W(G, H) I = I W(G, K) I + I W(G, L) I ' by 1.44. But G is connected, by 10.12, since n > l by definition. We deduce that 1 < I W(G, K) I < n and 1 < I W(G, L) I < n. We must now have cx1(K) < I W(G, K) I , for otherwise K E QlG) for some k < n. Similarly, cxi(L) < I W(G, L) j . Hence, cx1(H)

= cx1(K) + cr.1(L) < I W(G, K) I + I W(G, L) I -

1

=n

- l.

But this is contrary to the definition of H. The theorem follows. 10.14. If cr.i(G) 10.12). 10.15.

> 2 and ,\(G) > 2, then G has no loop or isthmus (by 9.14 and

If cr.i(G) > 4 and ,\(G) > 3, then G is simple.

Proof. Suppose G is not simple. It has no loop, by 10.14. Hence it must have two links A and B with the same ends. Write H = G · {A, B}. Evidently HE QlG) for some k < 2. Hence ,\(G) < 2, contrary to hypothesis. 10.16. Suppose cr.1(G) > 2. Suppose further that G has a vertex x satisfying val(G, x) < ½cr.1(G). Then ,\(G) < val(G, x).

Proof. Let U be the set of all edges of G incident with x. Allowing for the possibility of loops, we write I U I I U I . Moreover, I W(G, H) I does not exceed the number of links of G belonging to U, and therefore I W(G, H) I < I U I . We note also that cr.1(H) > 0, even when U is null, since cr.1(G) > 2. Hence His a non-null proper subgraph of G. It follows that HE Qt(G) for some k < I U I• The theorem follows. In the remainder of this section we determine all the graphs of infinite connectivity. It is convenient to define a bond as a graph consisting solely of two distinct vertices and one or more links joining them. A bond with exactly n links is an n-bond. Thus the I-bonds are the link-graphs and the 2-bonds are the lunes.

-

THREE-CONNECTION

10.17. Let m be the greatest integer not exceeding ½cx1{G). Then 1'.(G) and only if G is (m + !)-connected.

99

= oo if

Proof. If Ji.(G) = oo, then G is (m + !)-connected by definition. Conversely, suppose that G is (m + !)-connected. Then there is no integer k such that G is k-separated, since condition (i) cannot be satisfied if k > fcxi{ G). Hence Ji.(G) = oo. 10.18. A( G) = oo if and only if G is a null graph, a vertex-graph, a loop-graph, a link-graph, a lune, a triangle, or a 3-bond.

Proof. Suppose first that cx1{ G) < 1. Then Ji.( G) = oo if and only if G is connected, by 10.17. This condition is fulfilled if and only if G is null, a vertexgraph, a link-graph, or a loop-graph. Suppose next that 2 < cx1{ G) < 3. Then Ji.( G) = oo if and only if G is nonseparable, by I 0.17. This condition holds if and only if G is a lune, a triangle, or a 3-bond, by §9.2. Consider the remaining case, in which 01:i{G) > 4. Suppose Ji.(G) = oo. Then G is simple, by 10.15. This implies that cx0{G) > 4. Hence G has a vertex x satisfying val(G, x) < fcx 1(G), by 1.21. Accordingly, Ji.(G) is finite by 10.16.

10.2. Flags. We define a flag of a graph Gas a minimal member of Q.,,(G). If G is 2-connected, its flags are connected graphs, by 10.13. 10.21. Let H be a proper subgraph of a 3-connected graph G, or of a flag F of a 2-connected graph G. Suppose further that cx1(H) > I, cx1(H) > 2, and I W(G, H) I < 2. Then His a link-graph and W(G, H) = V(H).

Proof. Since H ¢ QlG), we have cx1{H) = 1. Write E(H) = {A}. The ends of A are vertices of attachment of Hin G, by 9.141. Any other vertex of H would be isolated in G, since A is a link by 9.14 and I W(G, H) I < 2. This is impossible, since G is connected. The theorem follows. 10.22. If 1'.(G)

= 2, then G has two flags with no common edge.

Proof. There exists He Q 2( G). Moreover, H e Q2{ G), by 10.11. Each of the subgraphs Hand H contains a flag of G, by 2.11. 10.23. Suppose that Ji.(G) = 2. Let F be a separable flag of G, with vertices of attachment x and y. Then Fis an arc of length 2, with ends x and y.

Proof. Since Fis connected, we have Ji.(F) = 1. Choose Ke Q1 (F), and let K 1 be its complement in F. Let z be the vertex of attachment of Kin F. Suppose that {x, y} !: V(K). Then W(G, KJ = {z}, by 1.43. Hence G is I-separated, contrary to hypothesis. A similar argument applies if {x, y} !: V(KJ, by 10.11. We may now assume, without loss of generality, that x e V(K) - {z} and

100

CONNECTIVITY IN GRAPHS

y E V(KJ - {z}. This implies that W(G, K) = {x, z}, by 1.43. Hence K is a link-graph with vertices x and z, by 10.21. Similarly, K1 is a link-graph with ends y and z. The theorem follows. 10.24. Suppose that .\(G) = 2. Let F 1 and Fa be distinct flags of G having a common edge A. Then G has a simple or circular path P = (a0 , Ai, a1, A2 , a 2, A 3 ,a3 ),orcontainsa3-bondT = G · {A 1, A2, A 3 },such that Aa = A, G · {Ai,A 2 } = Fi, and G · {A 2, A 3} = F2, Proof. We denote the vertices of attachment of F; by X; and Y; (i = l, 2). The two flags are connected graphs, by 10. 13, and neither one is a subgraph of the other. Hence each of the graphs F 1 n F2 and F1 n F 2 has an edge. Case I: {xi, Yi} = {xa, Y2} By 1.42 and 1.44 we have W(G, F 1 n FJ s;; {x1 , Yi}, Hence A is a link with ends x 1 and Yi, by 10.21. This implies that W(G, (F1)~) s;; {xi, YJ}. Hence (FJ' is a link-graph G · {B} with vertices x 1 and Yi, by 10.21, and therefore F 1 = G · {B, A}. Similarly, there is a link C of G, with ends x 1 and Yi, such that F 2 = G · {A, C}. Thus the theorem holds with T = G · {B, A, C}. Case II: {xa, Y2} s;; V(F1) By 1.42 and 1.44 we find that W(G, F 1 n FJ and W(G, F 1 n FJ are subsets of {xi, YJ). Hence F 1 n F 2 and F 1 n F2 are link-graphs with the same ends x 1 and Yi, by 10.21. Accordingly, x 1 and Yi are the two vertices of attachment of Fa, Case II thus reduces to case I. Case III: {xa, y 2} s;; V(FJ We now find that W(G, F1 n FJ and W(G. F1 n FJ are subsets of {xi, Yi}, Hence F1 n F2 is a link-graph G · {B}, with vertices x 1 and Yi, by 10.21. We deduce that F1 n F2 has at least one edge, since F 1 E Qz(G). It follows, since .\(G) = 2, that F 1 n Fa has at least two vertices of attachment. Thus W(G, F1 n FJ = {xi, Yi}, Now F1 n F2 is connected, by 10.13. Hence each of x 1 and Yi is incident with an edge of F1 n F2 , as well as with the edge B of F 2 • Accordingly, x 1 and y 1 are the two vertices of attachment of F 2, and case III reduces to case I. Case IV. The remaining case We adjust the notation so that x 2 E V(F1) - V(F1) and y 2 E V(FJ - V(FJ. We may further suppose that x 1 E V(FJ - V(FJ and Yi E V(FJ - V(FJ. For if this cannot be arranged, we can reduce to case II or III by interchanging F 1 and F 2 • Using 1.42 and 1.44, we find that W(G, F 1 n FJ s;; {x2, x 1 }, W(G, F 1 n FJ s;; {xa, Yi}, and W(G, F1 n FJ s;; {y2, x 1}. Hence, by 10.21, F 1 n F 2 is a link-graph G · {A} with vertices Xa and Xi, F 1 n Fa is a link-graph G · {B} with vertices x 2 and Yi, and F 1 n F 2 is a link-graph G · {C} with vertices Ya and x 1 • Thus the theorem holds with P = (yi, B, x 2, A, x1 , C, yJ.

THREE-CONNECTION

101

10.3. Connectivity and symmetry. In this section we describe two vertices x and y of a graph Gas equivalent if there is an automorphism/ of G such that fx = y. Evidently this is a true equivalence relation. We refer to the corresponding equivalence classes of vertices of Gas the transitivity classes of V( G). 10.3 I. Let G be a connected simple graph in which every vertex is at least divalent. Then if G is transitive on its vertices, it is 2-connected. If instead V(G) has exactly two transitivity classes, then either ,\(G) > 2 or y(G) = 3. Proof. Suppose G is not 2-connected. Then, by 9.58 and 10.12, G has a terminal cyclic element H. The vertex x of attachment of H is incident with some edge A of H, by 9.31. The other end, y say, of A is in V(H) - {x}. Since y is at least divalent, and G is simple, y is adjacent to a second vertex z of V(H) - {x}. Now xis the only vertex of H that is a cut-vertex of G, by 9.41. Hence it is not equivalent to any other vertex of H. The first part of the theorem follows. If VG) has exactly two transitivity classes U1 and U2 , we may write x E U1 and V(H) - {x} £ U2 • Since z is equivalent toy, it.is adjacent to a member of Ui, and therefore to x. Thus G has a triangle with vertices x, y, and z. Hence G, being simple, has girth 3. 10.32. Let G be a graph, of connectivity 2, in which the valency of each vertex is at least 3. Let F be a.flag of G with vertices ofattachment x and y. Then neither x nor y is equivalent to any vertex of V(F) - {x, y}. Proof. Suppose x or y is equivalent to z E V(F) - {x, y}. Then z is a vertex of attachment of a flag F' of G. Since F' is connected, by 10.13, it has an edge A incident with z, and A is a common edge of F and F'. But then z is divalent in G, contrary to hypothesis, by 10.24. 10.33. Let G be a simple graph, of connectivity >2, in which the valency of each vertex is at least 3. Then if G is transitive on its vertices, it is 3-connected. If instead V(G) has exactly two transitivity classes, then either ,\(G) > 3 or y(G) 4 can be 4-connected; each triangle of such a graph belongs to Qa(G).

10.35. Let G be an m-cage, where 3

< m < 8. Then G is 3-connected.

Proof. The definition of an m-cage G implies that G is connected. For any graph of girth m and valency 3 must have a component with the same properties. By the results of Chapter 8 each of the m-cages under consideration, except the 7-cage, is transitive on its vertices. Moreover, the 7-cage has just two transitivity classes of vertices: one consists of the middle vertices and the other of the vertices of the inner and outer octagons. The theorem now follows by 10.31 and 10.33.

A wheel of order n, where n > 3, is the graph obtained from an n-gon P n by adjoining one new vertex h and n new links joining h to the n vertices of P n• We denote this wheel by Wn. It is clearly a simple graph. We refer to Pn as the rim and to h as the hub of Wn. The edges of Wn incident with h are the spokes. All wheels of the same order n are isomorphic, by 6.45. The wheels of orders 3, 4, and 5 are shown in Figure 10.3A.

FIGURE

10.JA.

We note that val(Wn, h) = n and that every other vertex of Wn is trivalent. W3 is the 4-clique and any one of its vertices may be specified as the hub. If n > 4 the hub of Wn is uniquely determined as the only vertex whose valency exceeds 3. Any automorphism of Wn leaves h invariant and so induces an automorphism of P n· On the other hand, any automorphism f of P n can

103

THREE-CONNECTION

be extended as an automorphism of Wn by writing j7t = h. Thus A( Wn) is isomorphic to A(Pn) and is thus a dihedral group of order 2n, by 7.2. We note that V(Wn) has just two transitivity classes, namely {h} and V(Pn). 10.36. Every wheel is 3-connected. Proof. The wheel Wn of order n is the union of its rim Pn and n triangles, each of these triangles having the hub h as one vertex and an edge of P n as one edge. Hence Wn is non-separable, that is 2-connected, by 9.15 and 9.17. Suppose Wn is not 3-connected. Then it has a connected flag F, by 10.13 and 10.22. Let the two vertices of attachment of Fbe x and y. Since Fis simple and a.1(F) > 2, the set V(F) - {x, y} is non-null. Hence V(F) - {x, y} = {h}, by 10.32. It follows that h can be adjacent in Wn only to x arid y . But this is contrary to the definition of Wn. 10.4. Operations on vertices and edges.

10.41. Let x be a vertex of a graph G. Let K be the subgraph of G generated by V(G) - {x}. Then either ,\(K) = ,\(G) = oo or ,\(K) > ,\(G) - I. Proof. Suppose the theorem fails. Then ,\(K) is an integer k not exceeding ,\(G) - 2. Choose HE QtCK), and let H 1 be the complement of Hin K. Then H1 E Q,.(K), by 10.11. Let S and S1 be the sets of all links of G joining x to vertices of V(H) V(H1) and V(HJ - V(H) respectively. Since neither H nor H 1 is in QtCG), these sets of links are both non-null. Let J be the subgraph of G obtained from H by adjoining the vertex x and the edges of S . Since S and S1 are non-null, we have W(G, J)

=

W(K, H) v {x},

by 1.43, and therefore J E Qk+ i(G). But then k supposition.

+

1

> ,\(G),

contrary to

10.42. Let A bean edge of a graph G. Then either ,\(G) = oo or,\(G.A)

> ,\(G)-1.

Proof. Suppose ,\(GA) is an integer k not exceeding ,\(G) - 2. Choose HE QiG.A), and let H 1 be its complement in GA. Then H 1 E QtCG~), by 10.11. Since neither H nor H v (G · {A}) can be in Q 1( G), each of them has exactly k + 1 vertices of attachment in G, by 9.61. Since neither subgraph is in Qk+1(G), we must have a.i{H) = k and a.1(H1) = k. Hence a.1(G) = 2k+ l < 2>..(G). It follows, by 10.17, that ,\(G) = oo.

10.43. Let A be an edge of a simple graph G, and let ,\(G~) be at least 3. Then ,\(G) > 3. Proof Assume that ,\(G)

=

k


2, and 1Xi{H1) > 1. It now follows, by 1.432 and 10.21, that H 1 is a link-graph G · {B} whose ends u and v are its vertices of attachment in GA. This implies that u and v are the two vertices of attachment of Hin G, and that H has no other vertices. We deduce that u and v are the ends of A, contrary to the hypothesis that G is simple. 10.44. Let A bean edge ofagr.aph G. Theneither.\(G) = ooor.\(G~) ;>.\(G) - 1. Proof Suppose .\(G~) is an integer k not exceeding ,\(G) - 2. Choose J E QiG~) and let J1 be its complement in G~. Then 1%1(/,. - 11) > k + I and W(G, f,.- 1J) < k + I, by 9.62. Since .\(G) > k + 1, it follows that IX1{J1) = k. Similarly, 1X1(J) = k, by 10.11. It follows, by 10.17, that.\(G) = oo.

10.45. Let A be an edge of a simple graph G. Let m be an integer such that val(G, x) > m < .\(G~) for each end x of A. Then .\(G) > m. Proof Write .\(G) = k and assume that k < m. Choose HE QiG). We may suppose that A E E(H), by 10.11. Now 1%1(/,.H) > k - I and I W(G~,J,. H) I < k, by 6.25. Since .\(G~)> k, it follows that 1X1(H) = k and I W(G~,J,. H) I = k. The latter result implies that A has one end x that is not in W(G, H), by 6.25. We thus have k = 1X1 (H) > val(G, x) > m > k. This contradiction establishes the theorem.

10.5. ~otial edges. In this section we seek a theorem for 3-connected graphs that is analogous to 9.65. Our theory is based on that of reference 1. An edge A of a 3-connected graph G is essential if neither GA nor G~ is 3-connected. 10.51. If a 3-connected graph G has an essential edge A, then 1X0(G) 1X1(G) > 6.

> 4 and

Proof Suppose that 1X1(G) < 4. One of GA and G~ is 2-connected, by 9.65, and therefore 3-connected, by 10.17. Hence A is not essential, contrary to hypothesis. We deduce that 1Xi{G) > 4. Then G is simple, by 10.15, and therefore 1X0(G) > 4. Moreover, each vertex is at least trivalent, by 10.16. Hence 1Xi{G) > 6, by 1.21. Let x be a vertex of a graph G incident with three distinct links, A, B, C.

THREE-CONNECTION

105

Then we call the subgraph G · {A, B, C} of Ga triad of G and say that xis its centre. 10.52. Let A be an essential edge of a 3-connected graph G. Then A is an edge either of a triangle or of a triad of G. Proof. There exists HE Q 2(G~), by 10.42, since ,\(G) is not infinite by 10.18 and 10.51. Let its complement in G~ be H 1 • Then Hand H 1 have just two common vertices u and v, and these are vertices of attachment of both Hand H 1 in G~, by 10.11. Since Q 2(G) is null, the edge A of G has one end x in V(H) - V(H1 ) and the other end yin V(H1 ) - V(H), by 9.61. (See Figure 10.5A.)

FIGURE

to.SA.

There exists J E QlG~, by 10.44. Let its complement in G~ be 11 • Since

QiG) is null, it follows from 9.62 that/A. - 1Jhasjust three vertices ofattachment in G, of which two are the ends x and y of A. Let the third be denoted by w. Let Kbe the complement off.. -1J in G, and let K1 be the subgraph obtained

fromfA. - 1Jby deleting the edge A. Then o:i{K) = o:1(JJ > 2 and o:i{K1) = o:i(J) > 2. Hence K and K1 must each have at least three vertices of attachment. It follows that W(G, K) = W(G, K1) = {x, y, w}. Each of these three vertices is incident with an edge of Kand an edge of K1 , by 10.13. (See Figure 10.5B.) Interchanging u with v, H with Hi, or K with K1 , as may be necessary, we can arrange that w -=I= u, w E V(H1 ), and u E V(K1). Consider the subgraph H n K. The vertices of attachment of H are u, v, and x. Of these, only v and x can be in V(K). Similarly, wand x are the only vertices of attachment of K that can be in V(H). Hence W(G, H n K) s;; {v, x}, by 1.44, since w is in V(H) only if w = v. But each edge of K incident with x belongs to E(H), and therefore o:i{H n K) > 1. Hence, by 10.21, H n K is a link-graph G · {B} with vertices x and v. We note the corollary that v E V(K). If w E V(H), that is, if w = v, we can repeat the above argument with H

106

CONNECTIVITY IN GRAPHS

and H 1 interchanged, finding that G has an edge C with ends y and v. Then A belongs to the triangle G · {A, B, C} of G. From now on, therefore, we may suppose that w ¢ V(H). Consider the subgraph H n K1 • The only vertices of attachment of H that can be in V(K1) are x and u; for, if v E V(K1), we have v E V(K) n V(K1) and therefore v = w, contrary to our supposition that w ¢ V(H). On the other hand, the only vertex of attachment of K1 that can be in V(H) is x. Hence W(G, H n K1) s; {x, u}. As E(H n K1) includes all the edges of K1 incident with x, it follows from 10.21 that H n K1 is a link-graph G · {D} with vertices x and u.

FIGURE

JO.SB.

Now any edge of G, other than A, that is incident with x belongs to E(H), and therefore either to E(H n K) or to E(H n K1). Hence the only such edges are Band D. Thus A belongs to the triad G · {A, B, D} of G, whose centre is x.

I

Let us use the symbol G(ab cd) to denote the following proposition: there are distinct vertices a, b, c, and d of G, and a non-null proper subgraph Hof G, such that a E V(H), b E V(H), and W(G, H) = W(G, H) = {c, d}.

10.53. Let F be a connected subgraph of a 3-connected graph G, and suppose F

has just three distinct vertices of attachment a, b, and c. Let u and v be members of V(F) - {a, b, c}, not necessarily distinct, such that F(ab I cu) and F(ac I bv). Then either a is monovalent in For b and c are adjacent in F.

Proof. There are subgraphs Hand K of F, with complements H 1 and K1 in

107

THREE-CONNECTION

Frespectively, such that a e V(H), b E V(HJ, W(F, H) = W(F, Hi) = {c, u}, a E V(K), c E V(KJ, and W(F, K) = W(F, Ki) = {b, v}. The two corresponding decompositions of Fare shown in Figure IO.SC.

a

FIGURE

JO.SC.

Suppose first that v E V(HJ. The vertices of attachment of Hin Fare c and u, and of these only u can belong to V(K). The vertices of attachment of Kin F are b and v. Of these, only v can belong to V(H) , and this can happen only if

v = u. Hence, by 1.44, W(F, H n K) s;; {u}, and so W(G, H n K) s;; {u, a}, by 1.43. Since a is not isolated in F, we have rxi(H n K) > I. It follows from 10.21 that H n K is a link-graph. Hence a is monovalent in F.

In the remaining case v E (V(H) - {u}, and therefore v =I=- u. The only vertex of attachment of Hi in F that can be in V(K) is u, and the only vertex of attachment of Kin F that can be in V(Hi) is b. Hence W(F, H 1 n K) s;; {b, u}, by 1.44, and therefore W(G, Hi n K) s;; {b, u}, by 1.43. But b e W(F, Ki), Hence b is incident with an edge of K, and this edge is in E(Hi n K) . Using 10.21, we deduce that Hi n K is a link-graph with vertices band u. We thus have u E V(K) - {v}. Another application of 1.44 and 1.43 now shows that W(F, Hi n Ki) s;; {b, c} and W(G, Hi n Ki) s;; {b, c}. But be W(F, K), and therefore b is incident with an edge D of Ki, Clearly DE E(Hi n KJ. Using 10.21 , we deduce that Hi n Ki is the link-graph G · {D}, with vertices band c. Accordingly b and c are adjacent in F. 10.54. Suppose that rxi(G) > 4 and .\(G) > 3. Let G ·.{A, B, C} be a triangle of G such that neither G~ nor G~ is 3-connected. Then A is an edge of some triad ofG.

108

CONNECTIVITY IN GRAPHS

Proof. Let the vertices of the triangle be a incident with B and C, b incident with C and A, and c incident with A and B. Let F be the graph obtained from G by deleting the edges A, B, and C. Then W(G,F) = {a, b, c}. Moreover, a:i(F) > 3, by 10.15 and 10.16. Hence F is connected, by 10.13. There exists HE QlG,.), by 10.18 and 10.42. Let its complement in be H 1 • Then H and Hi have just two common vertices u and w, these being vertices of attachment of both subgraphs in G,., by 10.11. Since Qi G) is null, the edge A may be supposed to have its end bin V(H) - V(HJ and its end c in V(Hi) - V(H). We then have CE E(H) and BE E(HJ, and we may assume without loss of generality that w = a. We note that K = He and Ki = (H1)i are subgraphs of F whose union is F. They have no common edges, and their only common vertices are a and u. (See Figure 10.5D.)

G,.

b

u,

C

FIGURE

10.50.

We have val(K, b) > 1, by 10.16, and val (K, u) > 1 since u E W(G, Hi). Suppose, however, that a is isolated in K. Let L be the complement of [a] in K. Then W(G, L) = {b, u} and a:1(L) > 1. Hence bis the centre of a triad of G, by 10.21. Similarly, neither c nor u is isolated in Ki, and if a is isolated in K1 , then c is the centre of a triad of G. We deduce that either a is incident with edges of both Kand K1 or G has a triad whose centre is b or c. In the former alternative we have F(bc I au).

THREE-CONNECTION

109

Moreover, a is then at least tetravalent in G and so is not the centre of a triad ofG. By a similar argument, in which B replaces A, we find that either F(ac I bv) for some v e V(F) or G has a triad with centre a or c. Assume that A is not an edge of any triad of G. Combining the above results, we find that F(bc I au) and F(ac I bv). Hence c is monovalent in F, by 10.15 and 10.53. But then c is the centre of a triad of G, which is contrary to assumption. The theorem follows. 10.55. Suppose that ac1( G) > 4 and A(G) > 3. Let G · {A, B, C} be a triad of G, with centre x, such that neither GA nor Ga is 3-connected. Then A is an edge of some triangle of G. Proof. Let the ends of A, B, and C, other than x, be a, b, and c respectively. These three vertices are distinct, by 10.15. Let F be the graph obtained from G by deleting the vertex x and its incident edges. Then W(G, F) = {a, b, c}. Moreover, aci(F) > 3, by 10.16, and Fis connected, by 10.13. There exists J E Qi GA), by 10.44. Let its complement in GA be J 1• Since QiG) is null, f ,4. - 1J has just three vertices of attachment in G, two of which are the ends a and x of A, by 9.62. We denote the third by w. Let H be the complement off,4. - 1 J in G. Write H 1 = (f,4. - 1 J);.. Then ac1(H) > 2 and ac1(H1) > 2. Hence W(G, H) = W(G, HJ = {x, a, w} since A(G) > 3. We may assume that BE E(H). Then CE E(H1) . Clearly xis monovalent in both Hand H 1 • Let Kand K1 be the graphs derived from Hand H 1 respectively by suppressing the vertex x and its incident edge (Figure 10.SE).

FIGURE

10.SE.

110

CONNECTIVITY IN GRAPHS

Suppose that b = w. Then W(G, K) = {a, b} and ai{K) >I.Hence some triangle of G has A and Bas edges, by 10.21. Similarly, if c = w, some triangle of G has A and C as edges. In the remaining case, b E V(K) - V(KJ and c E V(KJ - V(K). We thus have F(bc I aw). Moreover, b and c are then not adjacent in G and so there is no triangle of G having both B and C as edges. By a similar argument, in which B replaces A, we find that either B and A are two edges of a triangle of G, Band Care two edges of such a triangle, or F(ac I ht) for some t E V(F). Assume that A is not an edge of any triangle of G. Combining the above results, we find that F(bc I aw) and F(ac I ht). Hence a and bare adjacent in G, by 10. I 6 and 10.53. But this contradicts our assumption. The theorem follows. There is a principle of duality in graph theory with respect to which 10.54 and 10.55 are corresponding theorems. This may be discussed in a later work. 10.56. Suppose that a 1(G) > 4 and .\(G) > 3. In order that every edge of G shall be essential it is necessary and sufficient that G shall be a wheel. Proof. Suppose first that G is a wheel Wn. Let A be any edge of G. At least one end of A is divalent in G~, and so .\(G~ < 2 by 10.16. On the other hand, A is an edge of at least one triangle of G, and therefore G~ is not simple. Hence .\(G~ < 2, by 10.15. Thus A is an essential edge of G. Conversely, suppose every edge of G is essential. Choose an edge A 0 with ends a0 and h. By 10.52 and 10.55 A 0 is an edge of a triangle G · {A 0 , B 0 , A1 } . We denote the third vertex of this triangle by a 1 , arranging that B0 is incident with a0 and a 1 , and A1 with a 1 and h. B0 belongs to a triad of G, by 10.54. We can adjust the notation so that a 1 is the centre of this triad. We can thus write the triad as G · {B0 , Ai, B1}, where

FIGURE

10.5.F.

THREE-CONNECTION

111

B1 is distinct from A 0 • Let the end of B1 other than a 1 be a 2• Since G is simple, by 10.15, a 2 is distinct from h, a0 , and a 1 • By 10.55 B1 is an edge of a triangle. We can adjust the notation so that the second edge of this triangle incident with a 1 is A 1 • We denote the third edge by A 2• Thus the ends of A 2 are a 2 and h. (See Figure IO.SF.) Let H 2 be the subgraph of G defined by the edges A0 , Ai, A2, B 0 , B1 and their incident vertices. Suppose that val(G, h) = 3. Then, by 10.16, W(G, HJ = {a0 , a 2} and cx1(HJ > I. HenceH 2 isalink-graph withendsa0 anda2 , by 10.21. Accordingly, G is a wheel of order 3. · In the remaining case G has a subgraph Hi, for some j > 2, with the following specifications:

(i) E(Hi) = {Ao, Ai, .. ., Ai, B0 , Bi, .. ., Bj-1}. (ii) V(Hi) = {h, a0 , Oi, •.• , ai}. (iii) The ends of A 1 are hand a 1; the ends of B 1 are a 1 and a 1+i· (iv) The 2j + l edges listed in (i) are all distinct, and the j + 2 vertices listed in (ii) are all distinct. (v) If 1

< i 4.

l, then val(G, a1)

= 3.

(vi) val(G, h)

(vii) j has the greatest value consistent with the above conditions (i}-(vi). By 10.54 Ai is an edge of a triad. The centre of this triad is ai, by (iii) and (vi). We can write the triad as G · {Bi_ 1 , Ai, Bi}, where Bi is distinct from all the edges listed in (i), by (iii) and (iv). We denote the other end of Bi by ai+iThen ai +1 -=I= h, since G is simple. Suppose that ai+i = a0 • Adjoining B1 to Hi we obtain a wheel K of order j + I, with hub h. By (v) we have W(G, K.) £ {a0 , h}. By 10.15 and 10.21 K has no edges. Hence K is a detached subgraph of the connected graph G, and therefore G is identical with the wheel K. In the remaining case ai+t is distinct from all the vertices listed in (ii), by (v). Since Bi is an edge of a triangle, by 10.55, it follows, by (v), that G has an edge Ai+ 1 joining hand ai+t• Adjoining Bi, ai+l> and Ai+t to Hi we obtain a subgraph Hi+i of G that satisfies conditions (i) to (vi) withj replaced by j + I. But this is contrary to (vii). The theorem follows. 10.6. The 3-coonected graphs. Suppose we are given a complete list Lm of the non-isomorphic 3-connected graphs of m edges, for some m > 3. We consider the problem of constructing the corresponding list Lm+i form + 1 edges. We define the following operations I and II, which may be applicable to a graph HeLm.

112

CONNECTIVITY IN GRAPHS

I. We adjoin a new edge A, whose ends are taken to be distinct non-adjacent vertices of H . II. Let x be a vertex of H that is at least tetravalent. We partition the set of edges of H incident with x into two disjoint classes U and V such that I U I > 2 < I VI . We replace x by two distinct new vertices u and v, the first postulated to be incident with the members of U only, and the second with the members of V only. The incidence relations of Hnot involving x are all retained. Finally we adjoin a new edge A with ends u and v. Suppose a graph G is constructed from HE Lm by operation I. Then H = G~ and G is 3-connected by l 0.43. If instead G is constructed from H by operation II we can identify H with GA. Then G is 3-connected by 10.45. Consider, on the other hand, a graph G that is 3-connected and has just m + l edges. If every edge is essential, G is a wheel, by 10.56. If G has a nonessential edge A, then either G~ or GA is isomorphic with a member of Lm. Hence a graph isomorphic with G can be constructed from a member of Lm by one of the operations I and II, by 10.15 and 10.16. To construct Lm+i, therefore, we first take note of the wheel of order ½(m + l) if mis an odd number greater than 3. For each HE Lm in turn we apply operations I and II in all possible ways not equivalent under the symmetry of H . Striking out duplicates from the set of all resulting graphs (including the wheel, if any), we obtain Lm+i· The 3-connected graphs of < 3 edges are the graphs of infinite connectivity. They are given by 10.18. We observe that neither of the operations I and II is applicable to either member of L 3 • Hence L 4 and L 5 are null, and L 6 consists solely of the wheel W3 • Neither of our operations can be applied to W3 • Hence L 1 is null and L 8 consists solely of the wheel W 4 • Operation I can be applied to W4 in essentially only one way, and operation II in just two different ways. The resulting three members of L9 are shown in Figure 10.6A. One of them is the Thomsen graph, or 4-cage.

FIGURE

10.6A.

THREE-CONNECTION

113

L 10 is found to have just 4 members, one of these being the 5-clique. These graphs are shown in Figure 10.6B.

FIGURE

10.6B.

BIBLIOGRAPHY 1. W. T. Tutte, A theory of 3-connected graphs, Nederl. Alcad. Wetensch. Proc., 64 (1961), 441-445.

11 THE STRUCTURE OF TWO-CONNECTED GRAPHS 11.1. Preliminaries. In the present chapter we construct a theory of 2connected graphs that is somewhat analogous to that given in §9.5 for the analysis of connected graphs into cyclic elements. We use the symbol [x, y] to denote an edgeless graph with just two vertices x andy. Suppose that [x, y] s;; G. We say G is 2-connected modulo [x, y] if the following two conditions hold:

(i) G is connected. (ii) For each z e V(G) each [z]-component of G has a vertex in {x, y} - {z}.

11.11. Let G be a 2-connected graph mod [x, y], where {x, y} s;; V(G). Then G is both [x]-connected and [y]-connected. Proof. Putting z = x in condition (ii) we find that each [x]-component of G has y as a vertex. Hence G has just one [x]-component, by 2.32. This [x]component has x as a vertex, by condition (i), and is therefore identical with G, by 2.45. Hence G is [x]-connected, by 2.43. Similarly, G is [y]-connected.

11.12. Let x and y be distinct vertices of a 2-connected graph G. Let H be an [x, y]-detached subgraph of G that is not contained in [x, y]. Then [x, y] s;; H, and H is 2-connected mod [x, y]. Proof. Let J be a subgraph of H that is either null or a vertex-graph [z]. Let C be any I-component of H. Then C must have a vertex in [x, y] - V(I), for otherwise C would be a proper I-detached subgraph of G not contained in I, and this would contradict the definition of G as 2-connected. Consider the special case I = n. There is a component D of H that is not a subgraph of [x, y]. Assume x ¢ V(D). Then D is a proper [y]-detached subgraph of G that is not contained in [y]. But this is impossible, since G is 2-connected. We deduce that x, and similarly y, is a vertex of D. Hence His connected, by 2.32 and the result of the preceding paragraph, and it has both x and y as vertices. The theorem follows.

114

THE STRUCTURE OF TWO-CONNECTED GRAPHS

115

11.13. Let x and y be distinct vertices of a graph G. Let Hand K be subgraphs of G such that H n K = [x, y]. Suppose further that H and K are both 2connected mod [x, y]. Then H V K is 2-connected. Proof. Suppose that z e V{H). Then K is a [z]-connected subgraph of G, by 2.47 and 11 .11 . Since His connected, it is the union of its [z]-components. Each of these is [z]-connected, by definition, and has either x or y as a vertex. Hence H v K is [z]-connected, by 2.32. Similarly, H v K is [t]-connected, for each t e V(K). Thus H v K is non-separable. It is thus 2-connected, by 10.12.

11.14. Let x and y be distinct vertices of a 2-connected graph G. Let H be an [x, y]-detached subgraph of G. Let z and t be vertices of G such that z e V(H) - {x, y}. Let K be a [z, t]-detached subgraph of G such that neither K nor its complenent K is a subgraph of [z, t]. Then either K c H, or K c H, or K and K are both I-separated. Proof. We may adjust the notation so that x =I= t. We may also arrange, interchanging Kand Kif necessary, that x e V(K) - V(K). By 11.12 Kand K both contain [z, t], and each is 2-connected mod [z, t]. Similarly, since z e V(H), the graph H contains [x, y] and is 2-connected mod [x, y].

FIGURE

11.lA.

Consider first the case t e V(H), illustrated in Figure 11.IA. Then W(G, K n H) £ {y}, by 1.42 and 1.44. But a1(H v K) > a.(K) > 1, since K is connected. Hence K n H is edgeless, by the 2-connection of G. Accordingly, E(K) £ E(H). Since K is connected and contains [z, t], it follows that K £ H . But x e V{H) - V(K). Hence K c H. We may now suppose that t e V(H) - {x, y}. Assume that ye V(K). Then y ¢ V(K) . Moreover, W(G, K n H) £ {z}, by 1.42 and 1.44. Similarly

116

CONNECTIVITY IN GRAPHS

W(G, K ri II)£ {t}. Hence G is not 2-connected, contrary to hypothesis. We deduce that in fact y E V(K) - V(K). It is now clear that (K ri H) ri (K ri H) is nor [x]. But K ri Hand K ri H have vertices z and t respectively that are distinct from x. Hence K, though connected, is not 2-connected. It is thus I-separated. A similar argument, in which K replaces Kand y replaces x, shows that K is I-separated.

11.2. A mapping of subgraphs. Let x and y be distinct vertices of a 2-connected graph G. Let H be an [x, y]-detached subgraph of G such that neither H nor H is contained in [x, y]. Then [x, y] = H ri H, and both Hand Hare 2-connected mod [x, y], by 11.12. Leta graph K be formed from H by adjoining a new edge A with ends x and y. Let S(K) denote the class of subgraphs of K, and let S(G, H) denote the class of thosesubgraphsLofGforwhicheither L £ Hor H £ L. We define a mapping gA of S(K) into S(G, H) as follows: (i) If N

£

H, thengA N

(ii) // N £ Kand A

E

= N.

E(N), then gA N

= (N~) u H.

Figure 11.2A shows the relations between G, H, H, and K.

K FIGURE ]

J.2A.

11.21. gA is a 1-l mapping of S(K) onto S(G, H). Proof. It is clear that gA maps distinct members of S(K) onto distinct members of S(G, H). Thus gA is a 1-1 mapping of S(K) onto a subset T of S(G, H). Suppose that YE S(G, H). Define XE S(K) as follows. If Y £ H, then X = Y, but if H £ Y, then X = (H ri Y) u (K · {A}). It is readily verified that gA X = Y. Hence T = S(G, H).

THE STRUCTURE OF TWO-CONNECTED GRAPHS

117

11.22. Let X and Y be subgraphs of K. Then g,. X £ g,. Y if and only if X £ Y. This theorem follows at once from the definition of g,.. 11.23. Let X and Y be subgraphs of K. Then g,.(X u Y) g,. (X n Y) = g,. X n g,. Y.

= g,.

Xu g,. Y and

Proof This theorem follows from the fact that, for each NE S(K), the graphs N and g,.N have identical intersections with H, together with the rule that His contained in g,. N if and only if K · {A} is contained in N.

11.24. Suppose that X £ Y £ K, and let Z be the complement of X in Y. Then g,. Z is the complement of g,. X in g,. Y. Moreover, W(g,. Y, g,. X) = W(Y, X). Proof g,. X has a complement Ting,. Y, by 11.22. Since g,. Xu g,. Z = g,.(X u Z) = g,. Y, by 11 .23, we have T £ g,. Z . g,. X n g,. Z = g,.(X n Z) = X n Z, by 11.23 and the fact that A is not an edge of both X and Z. By the definition of a complementary subgraph, X n Z is edgeless and V(X n Z) = W( Y, X) . Suppose vis a vertex of W( Y, X). It is incident in K with an edge B of E( Y) - E(X). If B-=/=- A, we have BE E(g,. Y) - E(g,. X). If B = A, we have v = x or y, X £ H, and g,. X = X..Moreover, His then a subgraph of g,. Y and therefore v is incident in g,. Y with an edge of H. In either case we have v E W(g,. Y, g,. X) . Since g,. Z £ g,. Y, by 11.22, it follows that g,. Z £ T. Combining the above results, we find that T = g,. Zand that W(g,. Y, g,. X) = V((g,. X) n T) = V(X n Z) = W( Y, X) .

11.25. Let J and X be subgraphs of K such that J £ X n H and H is a Jconnected subgraph of G. Then g,. maps the class of all J-components of X onto the class of all J-components of g,. X . Proof By 11.24 g,. maps the class of all J-detached subgraphs of X onto the class of all J-detached subgraphs of g,. X contained in S(G, H). Let T be any J-detached subgraph of g,. X. Suppose Tis not contained in H. Then we have A E E(X) and H £ g,. X. Moreover, by 2.54, T contains a J-component C of g,. X such that C n His not a subgraph of[x, y] and so not a subgraph of J. But H is connected mod H n J, that is it is a J-connected subgraph of g,. X by 2.47. Hence H £ C £ T, by 2.32. We deduce that TE S(G, H) in every case. The theorem now follows from 11.22, by §2.4.

11.26. Let J and X be subgraphs of K such that [x, y] £ J £ X n H . Then the J-components of X other than K · {A} are the J-components ofg,. X not contained in H. I

118

CONNECTIVITY IN GRAPHS

Proof. Let C be a J-component of X that is not K · {A}. Then A rf. E ( C), for otherwise K · {A} would be a proper subgraph of C, J-detached in X, and not contained in J. Hence g.4. C = C, and C is J-detached in g.4. X by 11.24. Moreover, C is a J-connected subgraph of g.. X. It follows that C is a Jcomponent of g.4. X. Clearly C is not a subgraph of H. Now let D be a J-component of g.. X that is not a J-component of X. By 2.441 and the result of the preceding paragraph we have D n H £ J. Hence H £ g.4. X, and D n H is not a subgraph of J. But H is J-detached in g.4. X. Hence D £ H, by 2.44. 11.27. Let X be any subgraph of K. Then g.4. X is connected if and only if X is connected. This theorem is the special case J

=

Q of 11.25.

11.28. Let X be any subgraph of K other than K · {A}. Then g.4. Xis 2-connected

if and only if X is 2-connected.

Proof. We may suppose that A E E(X) and H £ g.. X since otherwise X and g.4. X are identical. Suppose X is not 2-connected. There is a subgraph J of X, either null or a vertex-graph, such that Xis not J-connected. But then J £ g.4. X, and g.. X is not J-connected by 11.11 and 11.25. Hence g.4. Xis not 2-connected. In the remaining case Xis 2-connected. Applying I 1.12 to the graphs G and X we find that Hand Xi are 2-connected mod [x, y]. But H n Xi = [x, y] andHv Xi= g.4.X.Henceg.. Xis2-connected,byll.13.

11.3. Hinges. Let x and y be distinct vertices of a 2-connected graph G. We say that an [x, y]-component C of G is excisable if the following two conditions are satisfied : (i) oci(C)

> 2 < oc1(C).

(ii) Either C or C is 2-connected. If an excisable [x, y]-component C of G exists, we say that [x, y] is a hinge of G. We also describe Casa hinge-component of G. Condition (i) implies that G has at least two [x, y]-components. A hinge [x, y] of G is of Type I if G has exactly two [x, y]-components. Both [x, y]components are then excisable. In the remaining case we say the hinge is of Type II. 11.31. Let [x, y] be any hinge of G. Then (!ach [x, y]-component of G contains [x, y](by 11.12). 11.32. Let [x, y] be a hinge ofG of Type II. Then an [x, y]-component C ofG is excisable if and only if oci(C) > 2.

THE STRUCTURE OF TWO-CONNECTED GRAPHS

119

Proof. The other [x, y]-components of G have each at least one edge, and their union is C, by 2.45 and 11.12. Hence oci( C) > 2. Moreover, C is 2connected, by 11.12 and ll.13. The theorem follows.

11.33. Let [x, y] and [z, t] be distinct hinges of G. Then there is an [x, y]-component C of G such that [z, t] £ C. Proof. We may adjust the notation so that z is not x or y. Let C be the [x, y]component of G such that z e V( C). Let K be an excisable [z, t]-component of G. Then K £ C or K £ C, by 11.14. Hence [z, t] £ C, by ll.12.

11.34. Let G be a 2-connected graph. In order that G shall have no hinge, it is necessary and sufficient for it to be either a 3-connected graph or else a polygon or bond with at least four edges. Proof. If G has a hinge [x, y], then conditions (i) and (ii) imply that G is 2-separated. Hence if G is 3-connected, it has no hinge. If G is a bond with vertices x and y, then each [x, y]-component of G has exactly one edge, and so [x, y] is not a hinge of G. Finally, if G is a polygon and if x and y are two distinct vertices of G, then the [x, y]-components of Gare the two residual arcs of x and yin G, by 3.61 and 3.73. If either of these has more than one edge, it is separable, by 9.14. Hence [x, y] is not a hinge of the polygon G. Conversely, suppose G has no hinge. Assume that G is not 3-connected. Then G is 2-separated. Accordingly, we can find H c G such that 2 < 3. Then each cleavage unit of G is either a 3connected graph with at least 3 edges, or else a polygon or bond with at least 4 edges. Each edge of G belongs to just one cleavage unit, and each virtual edge of G to exactly two. Proof. Let m be the number of virtual edges of G. If m = 0, the theorem is true, by 11.34. Assume that it is true whenever m is less than some positive integer q, and consider the case m = q. Choose a virtual edge A of G. Let the cleavage graphs of G at A be D1 and D 2 • Each has at least three edges, by ll.41, and less than q virtual edges. Accordingly, the theorem is true for the 2-connected graphs D1 and D 2 • It is therefore true for G, by 11.62. The theorem follows in general by induction. Polygons or bonds of 3 edges may occur as cleavage units: they are 3connected graphs. Figure 11.6A shows the cleavage units of the graph G of Figure 1I.SA. Given a 2-connected graph G with at least three edges we can define another graph T(G) as follows. The vertices of T(G) are the cleavage units of G, and the edges of T(G) are the virtual edges of G. If A is a virtual edge of G, then its two ends in T(G) are the two cleavage units to which it belongs.

11.64. T(G) is a tree. Proof. If G has no virtual edge, then T(G) is a vertex-graph and the theorem is trivially true.

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CONNECTIVITY IN GRAPHS

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Suppose, however, that G has a virtual edge A. Let the cleavage graphs of G at A be D 1 and D 2 • Suppose T(D 1) and T(D 2) are trees. We may regard them as the two components of a disconnected graph H. Then T( G) is formed from Hby adjoining the edge A, with one end in V(T(D 1)) and the other in V(T(D 2 )), by 11.62 and 11.63. Hence T( G) is a tree, by two applications of 3.36. The theorem follows by a simple induction. 11.65. Let G be a 2-connected graph having at least one hinge. Then there exist two distinct cleavage units of G each of which includes just one virtual edge of G (by 3.35 and 11.64). In order to analyze a given 2-connected graph into its cleavage units we need a procedure for determining excisable hinge-components. The cleavage units can then be obtained by repeated applications of 11.62. We point out one possible procedure for determining excisable hinge-components in the following chapter.

12 NODAL THREE-CONNECTION 12.1 Nodes and branches. We define a node of a graph G as any vertex of G that is not divalent in G, and we denote the set of all nodes of G by N(G). In this chapter we do not distinguish between the set N(G) and the edgeless subgraph J of G such that V(J) = N(G). Thus we speak of N{G)-detached and N(G)-connected subgraphs of G. We refer to the N{G)-components of Gas the branches of G, and we denote the set of all branches of G by B(G). We say that a node x and a branch X of Gare incident if x e V(X). This chapter is based on the theory of nodes and branches given in Reference 1. 12.11. Let X be any branch of a graph G. Then I V(X) n N(G) I < 2. If Xis incident with two distinct nodes x and y of G, then Xis an arc with ends x and y. In the remaining case Xis a polygon. Proof. Suppose Xis incident with two distinct nodes x and y of G. There is an arc L s:; X that joins x and y and avoids N( G), by 3.62. This arc is N( G)detached since each of its vertices other than x and y is divalent in both L and G. Hence X = L by 2.42 and the definition of an N(G)-component. It follows that x and y are the only members of V(X) n N(G). In the remaining case there is at most one vertex of X that is not divalent in X. Hence val(X, x) is even for each x e V(X), by 1.22. It follows that X contains a polygon P, by 4.34. But then Pis N(G)-detached in G since each vertex of V(P) - N(G) is divalent in both P and G. Hence X = P by the definition of an N(G)-component.

By 12.11 the number of nodes of G incident with a given branch Xis either 2, I, or 0. In the first case the branch is an arc joining t.h e two nodes. In the second case the branch is a polygon and the incident node is its single vertex of attachment. In the third case the branch is a polygon that is detached in G. Such a branch we call isolated. 12.12. The isolated branches of a graph Gare those components of G that are polygons. Proof. Let P be a component of G that is a polygon.Pis trivially N(G)-

127

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CONNECTIVITY IN GRAPHS

detached. It is connected and therefore N(G)-connected, by 2.47. Hence Pis a branch of G. It is an isolated branch since all of its vertices are divalent in G. Conversely, let P be an isolated branch of G. Then, as we have seen, Pis a detached polygon of G. Thus, since polygons are connected, Pis a component ofG. We say that nodes and branches of Gare the vertices and edges respectively of the branch-graph of G. The structure of the branch-graph is defined by the incidence relations between the nodes and branches of G. Thus if G has no isolated branches, that is, polygonal components, then the branch-graph of G is itself a true graph. Otherwise the branch-graph is an example of a somewhat · more general mathematical system. 12.13 Let G be a graph without divalent vertices. Then its branch-graph His a true graph. Moreover, there is an isomorphism/ of G onto H such thatfx = x for each x E V(G), and/A = G · {A}for each A E E(G). Proof. Since G has no divalent vertex, it has no detached polygon. Hence H is a graph. Moreover, we have N(G) = V(G). Hence the branches of Gare the subgraphs G · {A}, A E E(G). The theorem follows.

12.2. Nodal subgraphs. A subgraph Hof a graph G is called nodal if it has the following property: if x is a vertex of H that is divalent in G, and if Xis the branch of G having x as a vertex, then X £ H. We note that Xis uniquely determined by x, by 2.441 and 2.45. 12.21. A subgraph Hof G is nodal if and only

if

W(G, H) £ N(G).

Proof. Suppose first that His nodal. Consider any x E V(H). If x ¢ N(G), then val(G, x) = 2 and xis a vertex of a branch X of G. In this case the two edges of G incident with x are both in E(X), by 2.441. As X £ H, by supposition, it follows that x ¢ W(G, H). Hence W(G, H) £ N(G). Conversely, suppose that W(G, H) £ N(G). Let x be any vertex of H that is divalent in G, and let X be the branch of G such that x E V(X) . Assume that there is a divalent vertex y of G that is in V(X) - V(H). There is an arc L £ X joining x and y and having no vertex in N(G), by 3.62. The graph H n L is a proper non-null subgraph of L , and therefore has a vertex of attachment z in L. But then z E W(G, H) £ N(G), by 1.43, and we have a contradiction. We deduce that each vertex of X not in N(G) belongs to V(H). But each edge of Xis incident with such a vertex, by 12.11 and the existence of x. Hence E(X) £ E(H) and therefore X £ H. We conclude that His a nodal subgraph of G.

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12.22. Any union or intersection of nodal subgraphs of G is a nodal subgraph of G (by 1.44 and 12.21). 12.23. The complement of a nodal subgraph of G is a nodal subgraph of G (by 1.51 and 12.21). 12.24. Each component of G is a nodal subgraph of G (by 12.21).

12.3. Nodal n-connection. Let k be a non-negative integer. We say that a graph G is nodally k-separated if it has a non-null proper nodal subgraph H with the following properties:

(i) Each of Hand H contains at least k distinct branches of G. (ii)

I W(G, H) I = k.

We denote the class of all such nodal subgraphs H by R,.( G). Thus if He R,.( G), we have also HE R,.(G), by 12.23. The nodal connectivity µ( G) of G is the least integer k, if such exists, such that G is nodally k-separated, that is, such that Rk( G) is non-null. If there is no such integer, k, we writeµ( G) = oo. We say that G is nodally n-connected, where n is a positive integer, if n < µ( G). 12.31.

If O< k
1, we have XE R1(G). But this is contrary to the hypothesis that G is nodally 2-connected. 12.36. Let G be a nodally 2-connected graph such that some branch X of G is a polygon. Then G = X. Proof. If Xis an isolated branch of G, we must have X = G, since otherwise XeRo(G). If Xis not isolated, then V(X) contains a single node x of G, and xis the only member of W(G, X). Hence XE R1( G), contrary to the hypothesis that G is nodally 2-connected.

We define a 8-graph as any graph G that is the union of three internally disjoint arcs, Li, L 2, and La, with common ends x and y. For such a graph it is clear that N(G) = {x, y} and B(G) = {Li, L 2, La}, 12.37. If G is a 8-graph, then G is non-separable. Proof. The union of any two branches of G is a polygon, by 3. 72. Hence G is non-separable, by 9.16 and 9.17.

12.38. If G is a 8-graph, then µ,(G) = oo. Proof. Any non-null proper nodal subgraph of G has just two vertices of attachment. Hence RiG) is null if O < k < l. But R,.(G) is null if k > 2, since G has only three branches.

12.4. Nodal 2-connection. The followingtheorem is often helpful in determining whether a given k-separated graph is also nodally k-separated. 12.41. Let H be a subgraph of a graph G. Let K be the union of all branches of G having an edge in common with H. Then I W(G, K) I < I W(G, H) I Proof. K is a nodal subgraph of G, and its vertices of attachment are nodes of G, by 12.21. For each x E W(G, K) we define a corresponding x' E W(G, H) as follows: if x E W(G, H), then x' = x. Suppose xis not in W(G, H). Then xis not in V(H), for since xis in W(G, K) it is incident with at least one edge of G not in E(H). It follows that x is incident in G with no member of E(H). We can find a branch X of G incident with x and such that E(X) n E(H) is not null, by the definition of K. If Xis an arc with ends x and y, we can find a simple path P in G from x to y such that G(P) = X, by 4.35. In the remaining case we can find a circular path Pin G, with origin and terminus x, such that G(P) = X, by 4.36 and 12.11. We define x' as the first vertex in P that is incident with an edge of H. Clearly x' is not the origin or terminus of P. Hence x' is a vertex of attachment of Hin G, and it is not a node of G.

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Suppose x and y are distinct vertices of W(G, K) such that x' = y'. Then it is clear from the above definitions that x' is not a node of G; it is an internal vertex of a branch X with distinct incident nodes x and y. By 4.35 there is just one simple path Pin X from x toy. It follows that x' is the first vertex incident with an edge of Hin both P and p- 1 • Accordingly, it is the only vertex of X incident with an edge of H. But this is impossible, since no edge of the arc X is a loop. We deduce that distinct vertices x and y of W(G, K) determine distinct vertices x' and y' of W(G, H). The theorem follows. 12.42. Let G be any graph that is not an arc. Then G is nodally 2-connected if and only if it is non-separable. Proof. If G is non-separable, it is nodally 2-connected, by 10.12 and 12.33. Conversely, suppose G is nodally 2-connected. Then G is connected, by 12.31 and 12.34. Assume that G is separable. There there is a vertex x of G such that G is not [x]-connected. If x E N(G), each [x]-component of G belongs to Ri(G), contrary to the supposition that G is nodally 2-connected, by 12.21. We may therefore suppose that val(G, x) = 2. Let X be the branch of G such that XE V(X). If X is a polygon, then G = X, by 12.36. But then G is non-separable, contrary to assumption, by 9.17. We may therefore suppose Xto be an arc. Let H be any [x]-component of G. Then H has no isolated vertex, by the connection of G. Let K be the union of all branches of G having an edge in E(H). If any branch Y of G has edges in both E(H) and E(H), we must have x E V( Y), for otherwise H n Y would be a detached non-null proper subgraph of Y and this is impossible since Y is connected. This implies that Y = X. We deduce that K = H u X. Now W(G, K) < 1, by 12.41. Hence K = G, for otherwise K would belong to ~(G) or Ri(G), contrary to the supposition that G is nodally 2connected. Since K = H u X, it follows that every [x]-component of G other than His a subgraph of X. But G has at least two [x]-components, and any one of them may be taken as H in the above reasoning. Hence G £ X , by 2.45, and so G = X . This result is contrary to the hypothesis that G is not an arc. We conclude that in fact G is non-separable.

I

I

12.S. Homeomorphisms. A homeomorphism f of a graph G onto a graph H is a pair {g, h}, where g is a 1-1 mapping of N(G) onto N(H) and his a 1-1 mapping of B(G) onto B(H), such that a node xis incident with a branch X in G if and only if gx is incident with hX in H. If such a homeomorphism exists, we say that G and Hare homeomorphic.

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CONNECTIVITY IN GRAPHS

A homeomorphism/= {g, h} of G onto H determines a homeomorphism J- 1 = {g- 1 , h- 1} of H onto G, called the inverse off. Moreover, if/1 = {gi, h1 } is a homeomorphism of H onto a graph K, then there is a homeomorphism Ji! = {g1 g, h1 h} of G onto K, called the product of / 1 and f in that order. We note too that there is a trivial homeomorphism, in which the mappings g and h reduce to identities, of G onto itself. Being homeomorphic is thus an equivalence relation for graphs. It arranges graphs in disjoint homeomorphism classes such that two graphs are homeomorphic if and only if they belong to the same homeomorphism class. Let/ = {g, h} be a homeomorphism of a graph G onto a graph H. Let K be a nodal subgraph of G. We can define a nodal subgraph/K of H as follows. First we take the union U of those branches of H that are images under h of branches of G contained in K. Then we adjoin to U those nodes of H, not already in V( U), that are images under g of nodes of G in V(K). The resulting graph isfK. Similarly, given a nodal subgraph J of H we can derive a nodal subgraphJ- 1J of G. From this definition we readily derive the following five theorems. 12.51. If Kand J are nodal subgraphs of G and H respectively, thenJ- 1(/K) = K andJ(f- 1J) = J. 12.52. f induces a 1-1 correspondence K +--+ f K between the nodal subgraphs K of G and the nodal subgraphs of H (by 12.51). 12.53. If Kand Lare nodal subgraphs of G such that Ks;; L, thenfK s;; JL. 12.54. If K is a nodal subgraph of G and x = val(JK, gx).

E

N(G) n V(K), then val(K, x)

12.55. If Kand Lare nodal subgraphs of G such that Ks;; L, then g maps W(L, K) onto W(JL,JK) (by 12.21 and 1.432).

It follows from 12.55 that/ induces a 1-1 mapping of Rt(G) onto Rt(H) for each non-negative integer k. Hence 12.56. Homeomorphic graphs have equal nodal connectivities. 12.57. Letf = {g, h} be a homeomorphism of G onto H . Let K be a nodal subgraph of G. Then K andfK are homeomorphic. Proof. By the definition of a nodal subgraph the nodes of Kare those nodes of G that are in V(K) and whose valency in K is not 2. A similar result holds forfK. Henceg induces a 1-1 mapping of N(K) onto N(JK), by 12.54. Let X be any branch of K. Each branch of G is N(K)-connected, by 2.47. Hence Xis a nodal subgraph of G, by 2.51. Accordingly, the nodal subgraph f X of His defined. It is a subgraph of fK, by 12.53. It is N(fK)-detached in fK, by 12.55. Hence there is a branch Y of fK such that Y s;; f X . Similarly,

NODAL THREE-CONNECTION

133

there is a branch Z of K such that Z