Splitting and contractible edges in 4-connected graphs

http://www.elsevier.com/locate/jctb

Journal of Combinatorial Theory, Series B 88 (2003) 227–235

Splitting and contractible edgesin 4-connected graphs

Akira Saito1

Department of Applied Mathematics, Nihon University, Sakurajosui 3-25-40, Setagaya-Ku,

Tokyo 156-8550, Japan

Received 20 November 2001

Abstract

Let G be a graph and let x be a vertex of degree four with NGðxÞ ¼ fa; b; c; dg: Then the

operation of deleting x and adding the edges ab and cd is called splitting at x: An edge e of a

graph G is said to be k-contractible if contraction of e yields a k-connected graph. Splitting has

been studied as a reduction method to preserve edge-connectivity. In this paper, we consider

splitting and 4-contractible edges as tools for reduction of 4-connected graphs. We prove that

for a 4-connected graph G of order at least six, there exists either a 4-contractible edge or a

vertex eligible for splitting preserving 4-connectedness near every vertex in G:r 2002 Elsevier Science (USA). All rights reserved.

Keywords: 4-Connected graphs; Contraction; Splitting

1. Introduction

For a graph G and an edge e of G; we denote by G=e the graph obtained from G bycontraction of e: If G=e is k-connected, then e is said to be a k-contractible edge.Since contraction of a k-contractible edge in a k-connected graph can be a tool forinductive arguments, the distribution of k-contractible edges can be usefulinformation. Tutte [12] proved that every 3-connected graph of order at leastfive contains a 3-contractible edge. Later, the distribution of 3-contractible edges in a3-connected graph has been well studied (see [5]).

E-mail address: [email protected] by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C),

13640138, 2001.

0095-8956/02/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved.

PII: S 0 0 9 5 - 8 9 5 6 ( 0 2 ) 0 0 0 4 3 - 6

In [1,3], it is proved that a 3-contractible edge exists near every vertex in a3-connected graph.

Theorem A (Halin [3], Ando et al. [1]). Let G be a 3-connected graph of order at least

five and let x be a vertex in G.

(1) If degG x ¼ 3; then a 3-contractible edge is incident with x.(2) If degG xX4; then either a 3-contractible edge is incident with x, or x has

three neighbors y1; y2; y3 such that a 3-contractible edge is incident with each yi

ði ¼ 1; 2; 3Þ:

Using Theorem A, Plummer and Toft [10] have given an upper bound to the cyclicchromatic number of a 3-connected planar graph.

For 4-connected graphs, the situation is completely different. Thomassen [11]remarked that there exist infinitely many 4-connected graphs which containno 4-contractible edges. Such graphs are called 4-contraction critical graphs. Later,4-contraction critical graphs have been characterized [8,9].

Theorem B (Martinov [8,9]). A graph G is 4-contraction critical if and only if either G

is the square of a cycle of length at least five, or G is the line graph of a cyclically

4-edge-connected cubic graph.

Since even the existence of 4-contractible edges is not guaranteed for 4-connectedgraphs, there seems to be no hope in studying their distribution. However, if weintroduce another reduction, the situation may change. We may be able to obtaineither distributive information on 4-contractible edges or a local structure that iseligible for the alternative reduction.

In this paper, we choose splitting as the alternative reduction. Since we applysplitting only to vertices of degree four and we consider vertex-connectivity, we use theterm ‘‘splitting’’ in a very restricted sense. Although the definition of ‘‘splitting’’ fits wellin the arguments of this paper, we remark that splitting usually has a more generaldefinition (see for example [4,6,13]). Let G be a graph and let x be a vertex of G ofdegree four. Let NGðxÞ ¼ fx1; x2; x3; x4g: Then we consider the following operation.

(1) delete the vertex x;(2) add the edge x1x2 if x1 and x2 are not already joined by an edge, and(3) add the edge x3x4 if x3 and x4 are not already joined by an edge.

We call this operation splitting at x; and denote the resulting graph by Gxx1;xx2:

In other words, Gxx1;xx2is the graph defined by VðGxx1;xx2

Þ ¼ VðGÞ � fxg and

EðGxx1;xx2Þ ¼ EðG � xÞ,fx1x2; x3x4g:

Splitting has appeared as a tool of reduction in many areas of graph theory such asinteger flows, cycle covers, graph coloring and connectivity. However, in the study ofconnectivity, splitting has only been considered in relation to edge-connectivity. Incontrast, the purpose of the paper is to claim that splitting can be a tool for reductionin relation to vertex-connectivity for 4-connected graphs. For example, the followingcorollary follows from Theorem B.

A. Saito / Journal of Combinatorial Theory, Series B 88 (2003) 227–235228

Corollary 1. Let G be a 4-connected graph of order at least six. Then either G has a

4-contractible edge, or G has a vertex x of degree four such that some splitting at x

yields a 4-connected graph.

The proof is done by investigating the local structure of 4-contraction criticalgraphs, which are characterized in Theorem B (note that all of these graphs are4-regular). We give its proof in Section 4, where we prove a stronger statement.

Since the Corollary 1 guarantees the existence of either a 4-contractible edge or avertex eligible for splitting, it may be worthwhile to study the distribution. Note that,in the investigation of the distribution, we cannot assume that the graph is4-contraction critical since contraction may change the distribution of 4-contractibleedges or vertices eligible for splitting.

In this paper, we prove the following theorem, which is a generalization ofTheorem A for 4-contractible edges.

Theorem 2. Let G be a 4-connected graph of order at least six, and let x be a vertex in

G. Then one of the following holds:

(1) A 4-contractible edge is incident with x.(2) There exists a vertex y of degree four in NGðxÞ such that a 4-contractible edge is

incident with y.(3) There exists a vertex y of degree four in NGðxÞ such that for some pair of edges e1;

e2 incident with y, Ge1;e2 is 4-connected.

We give a proof of Theorem 2 in Section 2. We study the splitting of 4-contractioncritical graphs in Section 3.

For graph-theoretic terminology not defined in this paper, we refer the reader to[2]. Let G be a graph and let ACVðGÞ:We define the neighborhood of A; denoted byNGðAÞ; by

NGðAÞ ¼[

xAA

NGðxÞ � A:

We denote by G½A� the subgraph of G induced by A:Let G be a non-complete graph, and let S be a minimum cutset of G: A union of at

least one, but not all, of components of G � S is called a fragment associated with S:In other words, A is said to be a fragment associated with S if NGðAÞ ¼ S and

A,SiVðGÞ: For a fragment A associated with a cutset S; we write %A for VðGÞ �ðS,AÞ: Note that %A is also a fragment associated with S:

2. Proof of Theorem 2

Before we prove Theorem 2, we make two observations. Though they are simple,they are frequently used in the subsequent arguments.

A. Saito / Journal of Combinatorial Theory, Series B 88 (2003) 227–235 229

Lemma 3. Let G be a k-connected graph ðkX2Þ and let e ¼ xyAEðGÞ:

(1) If e is not k-contractible, then G has a cutset of order k which contains fx; yg:(2) If NGðxÞ � fyg induces a complete graph, then e is k-contractible.

Proof. The definition of a k-contractible edge immediately yields (1).In order to prove (2), assume e is not k-contractible. Then by (1), G has a cutset S

of order k with fx; ygCS: Let A be a fragment of G associated with S; then since G is

k-connected, A-ðNGðxÞ � fygÞa| and %A-ðNGðxÞ � fygÞa|: Let uAA-ðNGðxÞ �fygÞ and vA %A-ðNGðxÞ � fygÞ: Then uveEðGÞ; which contradicts the assumptionthat NGðxÞ � fyg induces a complete graph. &

The proof of Theorem 2 is decomposed into several lemmas. First, we investigatethe local structure around a vertex to which no 4-contractible edges are incident.

The next lemma essentially appears in [9, Proposition 1; 7, Lemma 3], but we giveits proof to make the paper self-contained. (Note that though Proposition 1 in [9]and Lemma 3 in [7] assume that the graph is 4-contraction critical, their proofs onlyuse the same weaker assumption as in Lemma 4.)

Lemma 4. Let G be a 4-connected graph of order at least six, and let xAVðGÞ: If no

edge of G incident with x is 4-contractible, then there exists a vertex yANGðxÞ such that

degG y ¼ 4 and NGðxÞ-NGðyÞa|:

Proof. By Lemma 3(1), for each vANGðxÞ there exists a cutset S of order four withfx; vgCS: Let A be a fragment associated with S: Choose ðv;S;AÞ so that jAj is assmall as possible. Since G is 4-connected and jSj ¼ 4; NGðxÞ-Aa|; sayuANGðxÞ-A: By the assumption, there exists a cutset T of order four withfx; ugCT : Let B be a fragment associated with T : Let

X1 ¼ ðS-BÞ,ðS-TÞ,ðA-TÞ;

X2 ¼ ðA-TÞ,ðS-TÞ,ðS- %BÞ;

X3 ¼ ðS- %BÞ,ðS-TÞ,ð %A-TÞ

and

X4 ¼ ð %A-TÞ,ðS-TÞ,ðS-BÞ:

Note that jX1j þ jX3j ¼ jX2j þ jX4j ¼ jSj þ jT j ¼ 8:

First, we claim %A-Ta|: Assume %A-T ¼ |: Since %Aa|; either %A-Ba| or%A- %Ba|: By symmetry, we may assume %A-Ba|: Then X4 is a cutset and hence

jX4jX4: However, since %A-T ¼ |; X4CS: This implies X4 ¼ S and S- %B ¼ |: SinceuAT � S; jX3j ¼ jS-T jp3: Thus, X3 cannot be a cutset, and hence %A- %B ¼ |:Then, %BCA: Since uAA � %B; %BiA: Since %B is a fragment associated with T ; thiscontradicts the minimality of A: Hence the claim follows.


Next, we claim S-Ba| and S- %Ba|: Assume S-B ¼ |: Then since %A-Ta|and A-Ta|; X1iT and X4iT : Therefore, neither X1 nor X4 is a cutset of G: This

implies A-B ¼ | and %A-B ¼ |: However, we now have B ¼ |; a contradiction.

Similarly, we have S- %Ba|:Assume A-Ba|: Then X1 is a cutset. On the other hand, since uAA � B;

A-BiA: Since fx; ugCX1; the minimality of A requires jX1jX5: This implies

jX3jp3 and hence %A- %B ¼ |: Moreover, since jT j ¼ 4 > 3XjX3j; we have jA-T j >jS- %Bj:

Now since A-Ba|;

jAj ¼ jA-Bj þ jA-T j þ jA- %Bj

> j %A- %Bj þ jS- %Bj þ jA- %Bj ¼ j %Bj:

Since %B is a fragment associated with T ; this again contradicts the minimality of jAj:Therefore, we have A-B ¼ |: By a similar argument, we also have A- %B ¼ |: Thus,we have ACT : This implies jAjp2:

If jAj ¼ 2; then S-T ¼ fxg and j %A-T j ¼ 1: Since jS-Bj þ jS- %Bj ¼ 3; by

symmetry, we may assume jS-Bj ¼ 1: Then jX4jp3 and hence %A-B ¼ |: Thisimplies jBj ¼ 1; which contradicts the minimality of A: Thus, we have A ¼ fug: ThenNGðuÞ ¼ S: This implies degG u ¼ 4 and vANGðxÞ-NGðuÞ: &

Lemma 5. Let G be a 4-connected graph of order at least six and let x be a vertex of G

with degG x ¼ 4; say NGðxÞ ¼ fx1; x2; x3; x4g: If x3x4AEðGÞ and neither xx1 nor xx2

is 4-contractible, then we have either x1x2AEðGÞ; fx1x3;x2x4gCEðGÞ or

fx1x4; x2x3gCEðGÞ:

Proof. By Lemma 3(1), there exists a cutset Si of order four with fx; xigCSi ði ¼1; 2Þ: Since x3x4AEðGÞ; G � Si has exactly two components, say Ai and Ai ði ¼ 1; 2Þ:We may assume x2AA1 and x1AA2: Then A1-fx3; x4ga| and A2-fx3; x4ga|:Since x3x4AEðGÞ; A1-fx3; x4g ¼ | and A2-fx3;x4g ¼ |: In particular,

NGðxÞ-A1-A2 ¼ NGðxÞ-A1-A2 ¼ NGðxÞ-A1-A2 ¼ |: Let

X1 ¼ ðS1-A2Þ,ðS1-S2Þ,ðA1-S2Þ;

X2 ¼ ðA1-S2Þ,ðS1-S2Þ,ðA2-S1Þ;

X3 ¼ ðA2-S1Þ,ðS1-S2Þ,ðA1-S2Þ

and

X4 ¼ ðA1-S2Þ,ðS1-S2Þ,ðA2-S1Þ:

We claim A1-S2a|: Assume A1-S2 ¼ |: Since jS1-S2jp3; jX3j þ jX4j ¼ jS1j þjS1-S2jp7: Hence jX3jp3 or jX4jp3: If jX3jp3; then X3 cannot be a cutset, and

hence A1-A2 ¼ |: Since A1a|; we have A1-A2a|: Then since X4 ¼ S1 � A2 is a

cutset, S1-A2 ¼ |: This implies A1-A2a|: Since NGðxÞ-A1-A2 ¼ |; S2 � fxgseparates G: This contradicts the assumption that G is 4-connected. If jX4jp3; then


we have A1-A2 ¼ | and A1-A2a|: This implies that X3CS1 � fx1g is a cutset,again a contradiction. Thus, the claim follows.

Assume S1-A2 ¼ |: Then since A1-S2a| and x2AA1-S2; jX2jp3 and jX3jp3:

These imply A1-A2 ¼ | and A1-A2 ¼ |: Now we have A2 ¼ |; a contradiction.

Thus, we have S1-A2a|:Suppose A1-A2a|: Since A1-A2-NGðxÞ ¼ |; X1 � fx1g separates G and hence

we have jX1 � fxgjX4; which implies jX1jX5: Since jX1j þ jX3j ¼ jS1j þ jS2j ¼ 8;

jX3jp3 and A1-A2 ¼ |: Since A1-S2a|; S1-S2a| and S1-A2a|; we have

jA1-S2j ¼ jS1-S2j ¼ jS1-A2j ¼ 1; and jS1-A2j ¼ jA1-S2j ¼ 2: Since

A1-A2-NGðxÞ ¼ | and jX2j ¼ 4; A1-A2 ¼ |: Hence jA2j ¼ 1: Since

NGðxÞ-A2a|; we have A2 ¼ fx3g or A2 ¼ fx4g: By symmetry, we may assume

A2 ¼ fx3g: Since x4AA1 and x3x4AEðGÞ; S2-A1 ¼ fx4g: Since jX4j ¼ 4 and

NGðxÞ-A1-A2 ¼ |; A1-A2 ¼ |: Then A1 ¼ fx4g: This implies x2x3AEðGÞ and

x1x4AEðGÞ: Therefore, the lemma follows if A1-A2a|:Next, suppose A1-A2a|: Then jX2jX5; jX4jp3 and A1-A2 ¼ |: Since

A1-S2a|; S1-S2a| and S1-A2a|; we have jA1-S2j ¼ jS1-S2j ¼ jS1-A2j ¼1 and jA1-S2j ¼ jS1-A2j ¼ 2: Since jX1j ¼ 4 and NGðxÞ-A1-A2 ¼ |; A1-A2 ¼|: Therefore, we have A2 ¼ fx1g; which implies x1x2AEðGÞ: Therefore, the lemma

follows if A1-A2a|:Now suppose A1-A2 ¼ A1-A2 ¼ |: Then A1CS2: If A1 ¼ fx2g; then we

immediately have x1x2AEðGÞ and the lemma follows. Therefore, we may assume

jA1j ¼ 2: This implies S1-S2 ¼ fxg and jA1-S2j ¼ 1: Since jX3j þ jX4j ¼ jS1j þjS1-S2j þ 2jA1-S2j ¼ 7; jX3jp3 or jX4jp3: If jX4jp3; then A1-A2 ¼ |: Further-more, jX4jp3 also implies S1-A2 ¼ fx1g: This implies x1x2AEðGÞ: Thus, we may

assume jX3jp3 and A1-A2 ¼ |: This implies jS1-A2j ¼ 1 and jS1-A2j ¼ 2: Since

NGðxÞ-A2a|; we have A2 ¼ fx3g or A2 ¼ fx4g: We may assume A2 ¼ fx3g: ThenA1-S2 ¼ fx4g: Since jX4j ¼ 4 and NGðxÞ-A2-A1 ¼ |; A1-A2 ¼ |: Hence A1 ¼fx4g: Therefore, we have x1x4AEðGÞ and x2x3AEðGÞ; and the lemma follows. &

Next, we describe the situation in which splitting does not yield a 4-connectedgraph in terms of a cutset.

Lemma 6. Let G be a 4-connected graph of order at least six, and let x be a vertex in G

with degG x ¼ 4; say NGðxÞ ¼ fx1; x2; x3; x4g: If Gxx1;xx2is not 4-connected, then G has

a cutset S of order four and a fragment A of G associated with S such that

(1) xAS;(2) fx1; x2g-Aa| and fx1; x2gCA,S; and

(3) fx3; x4g- %Aa| and fx3; x4gC %A,S:

Proof. By the definition of splitting, G � x is a spanning subgraph of Gxx1;xx2: Since

G is 4-connected but Gxx1;xx2is not 4-connected, Gxx1;xx2

has a minimum cutset S0 of


order three, which is also a minimum cutset of G � x: Let S ¼ S0,fxg: Then jSj ¼ 4and S is a minimum cutset of G: Let A be a fragment of Gxx1;xx2

associated with S0:

Then VðGÞ � ðS,AÞ ¼ VðG � xÞ � ðS0,AÞ ¼ VðGxx1;xx2Þ � ðS0,AÞa|: Hence A

is a fragment of G associated with S: Let %A ¼ VðGÞ � ðS,AÞ: Then NGðxÞ-Aa|and NGðxÞ- %Aa|:

Assume fx1; x2g-A ¼ fx1; x2g- %A ¼ |: Then we may assume x3AA and x4A %A:Then x3x4AEðGxx1;xx2

Þ: However, since A is also a fragment of Gxx1;xx2; this is

impossible. Therefore, we have fx1; x2g-Aa| or fx1; x2g- %Aa|: We may assume

x1AA: If x2A %A; then since x1x2AEðGxx1;xx2Þ; A cannot be a fragment of Gxx1;xx2

; acontradiction. Hence x2AS,A:

Since NGðxÞ- %Aa|; fx3; x4g- %Aa|: We may assume x3A %A: Then by the same

argument as above, we have x4AS, %A: &

The next lemma corresponds to the last part of the proof of Theorem 2.

Lemma 7. Let G be a 4-connected graph of order at least six, and let x be a vertex of

degree four, say NGðxÞ ¼ fx1; x2; x3; x4g: Suppose x1x2AEðGÞ and x3x4AEðGÞ; and

none of xx1; xx2; xx3 and xx4 are 4-contractible. Then either Gxx1;xx3or Gxx1;xx4

is 4-

connected.

Proof. Assume neither Gxx1;xx3nor Gxx1;xx4

is 4-connected. Then by Lemma 6, G has

a cutset Si of order four and a fragment Ai associated with Si ði ¼ 1; 2Þ such that

(1) xAS1; fx1; x3g-A1a|; fx1; x3gCS1,A1; fx2; x4g-A1a| and

fx2; x4gCS1,A1; and(2) xAS2; fx1; x4g-A2a|; fx1; x4gCS2,A2; fx2; x3g-A2a| and

fx2; x3gCS2,A2:

By symmetry, we may assume x1AA1: Then since x1x2AEðGÞ; x2eA1; which implies

x4AA1: Since x2AS1,A1; we have x2AS1: Since x3AS1,A1; x3x4AEðGÞ and

x4AA1; we have x3AS1:We consider two cases.

Case 1: x1AA2: Since x1x2AEðGÞ; x2eA2: Hence x3AA2 and x2AS2: Furthermore,since x3x4AEðGÞ and x4AS2,A2; we have x4AS2: Let

X1 ¼ ðS1-A2Þ,ðS1-S2Þ,ðA1-S2Þ;

X2 ¼ ðA1-S2Þ,ðS1-S2Þ,ðS1-A2Þ;

X3 ¼ ðS1-A2Þ,ðS1-S2Þ,ðA1-S2Þand

X4 ¼ ðA1-S2Þ,ðS1-S2Þ,ðS1-A2Þ:Since x1AA1-A2; we have jX1jX4: Since jX1j þ jX3j ¼ jS1j þ jS2j ¼ 8; jX3jp4: On

the other hand, NGðxÞ-A1-A2 ¼ |: This implies A1-A2 ¼ |: If S1-A2 ¼ |; thenX1 ¼ ðS1-S2Þ,ðA1-S2ÞiS2 since x4AA1-S2: This contradicts jX1jX4: Hence


S1-A2a|: This implies jS1-A2j ¼ 1; S1-S2 ¼ fx; x2g; and S1-A2 ¼ fx3g:Similarly, if A1-S2 ¼ |; then X1iS1 and we have a contradiction. Therefore,

A1-S2a|; which implies A1-S2 ¼ fx4g and jA1-S2j ¼ 1: Now we have jX2j ¼ 4

and NGðxÞ-A1-A2 ¼ |; jX4j ¼ 4 and NGðxÞ-A1-A2 ¼ |: Therefore, A1-A2 ¼A1-A2 ¼ |:

At this stage, we have A1 ¼ fx4g and A2 ¼ fx3g: These imply NGðx4Þ ¼ S1 andNGðx3Þ ¼ S2: In particular, x2x3; x2x4AEðGÞ: Since fx2; x3; x4g induces K3; xx1 is 4-contractible by Lemma 3(2). This is a contradiction, and the lemma follows in this case.

Case 2: x1eA2: In this case, x4AA2: Since x3x4AEðGÞ; x3eA2; which implies

x2AA2: By the assumption of the case, x1AS2: Similarly, since x3AA2,S2 andx4AA2; we have x3AS2: Now by the same arguments as in Case 1, we can prove thatfx1; x2; x3g induces a K3 and xx4 is 4-contractible, a contradiction. &

Theorem 2 immediately follows from Lemmas 4, 5 and 7.

3. Splitting of 4-contraction critical graphs

Since the class of 4-contraction graphs is characterized in Theorem B, it is notdifficult to see the existence of a vertex eligible for splitting. Actually, we can give astronger statement.

Theorem 8. Let G be a 4-contraction critical graph of order at least six. Then for each

xAVðGÞ; there exist two edges e ¼ xa; xb incident with x such that Ge;f is 4-connected.

Moreover, if G is the square of a cycle, then we can take e; f so that Ge;f is also the

square of a cycle.

Proof. By Theorem B, G is either the square of a cycle or the line graph of acyclically 4-edge-connected graph. In either case, G is 4-regular and G½NGðxÞ� hastwo independent edges. Therefore, Lemma 7 guarantees the first part of the theorem.

Suppose G is the square of a cycle. Let C ¼ x1x2yxnx1 and G ¼ C2: ThenEðGÞ ¼ fxixiþ1; xixiþ2: 1pipng; where the suffices are counted modulo n: Since G isvertex-transitive, we may assume x ¼ xn: Then Gxnx2;xnxn�1

is the square of the cycle

x1?xn�1x1: &

Acknowledgments

The author is grateful to the referee for the helpful comments.

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Splitting and contractible edges in 4-connected graphs