View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by Elsevier - Publisher Connector ARTICLE IN PRESS

Journal of Combinatorial Theory, Series B 93 (2005) 23–32 http://www.elsevier.com/locate/jctb

Girth and treewidth

L. Sunil Chandrana,1, and C.R. Subramanianb a Max-Planck Institute fu¨r Informatik, Stuhlsatzenhausweg 85, 66123 Saarbru¨cken, Germany b The Institute of Mathematical Sciences, Chennai 600113, India

Received 17 April 2003 Available online 17 July 2004

Abstract

The length of the shortest cycle in a graph G is called the girth of G: In particular, we show 1 IðgÀ1Þ=2m that if G has girth at least g and average degree at least d; then twðGÞ¼Oðgþ1 ðd À 1Þ Þ: In viewof a famous conjecture regarding the existence of graphs withgirth g; minimum degree I m d and having at most cðd À 1Þ ðgÀ1Þ=2 vertices (for some constant c), this lower bound seems to be almost tight (but for a multiplicative factor of g þ 1). r 2004 Elsevier Inc. All rights reserved.

Keywords: Girth; Treewidth; Tree decompositions; Minors

1. Introduction

The notions of tree-decomposition and treewidth were first introduced (under different names) by Halin. (See [11, Chapter 12] for historical details.) Roughly speaking, the treewidth (pathwidth) of a graph G is the minimum k such that G can be decomposed into pieces forming a tree (path) structure with at most k þ 1 vertices per piece. Such a decomposition is called a tree (path) decomposition. More formally,

Corresponding author. Fax: 49-681-9325-599. E-mail addresses: [email protected] (L.S. Chandran), [email protected] (C.R. Subramanian). 1 Most of this research was done while L.S. Chandran was a Ph.D. student in the Indian Institute of Science and was supported in part by the Infosys Fellowship.

0095-8956/$ - see front matter r 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jctb.2004.05.004 ARTICLE IN PRESS

24 L.S. Chandran, C.R. Subramanian / Journal of Combinatorial Theory, Series B 93 (2005) 23–32

Definition 1. A tree decomposition of G ¼ðV; EÞ is defined to be a pair ðX; TÞ where X ¼fXi : iAIg is a collection of subsets of V (we call these subsets the nodes of the decomposition) and T ¼ðI; FÞ is a tree having the index set I as set of vertices such that the following conditions are satisfied: S 1. iAI Xi ¼ V: 2. 8ðu; vÞAE; (iAI : u; vAXi: 3. 8i; j; kAI:ifj is on a path in T from i to k; then Xi-XkDXj:

The width of a tree decomposition ðfXi : iAIg; TÞ is defined to be equal to maxiAI jXijÀ1: The treewidth of G is defined to be the minimum width over all tree decompositions of G and is denoted by twðGÞ: A path decomposition of G ¼ðV; EÞ is a tree decomposition ðX; TÞ in which T is required to be a path. The pathwidth of G is defined to be the minimum width over all path decompositions of G and is denoted by pwðGÞ:

These notions underly several important and sometimes very deep results in graph theory and graph algorithms and are very useful for the analysis of several practical problems. Recent research has shown that many NP-complete problems become polynomial or even linear time solvable, or belong to NC, when restricted to graphs with small treewidth (see [2–4]). The concepts of treewidth and pathwidth have applications in many practically important fields like VLSI layouts, Cholesky factorization, expert systems, evolution theory, and natural language processing (see [4] for references). The decision problem of checking whether twðGÞ is at most k; given G and k is known to be NP-complete. Hence, the problem of determining the treewidth of an arbitrary graph is NP-hard and the research on determining the treewidth and pathwidth has been focused on special classes. Linear or polynomial time or NC algorithms for producing optimal tree decompositions have been proposed for several special classes of graphs like graphs of bounded treewidth [5,6], chordal graphs, cographs, circular arc graphs, chordal bipartite graphs, permutation graphs, circle graphs, and distance hereditary graphs. For an extensive bibliography on treewidth, see [4]. The length of the shortest cycle in a graph G is called the girth of G: In this paper, we derive a lower bound for the treewidth of G in terms of its girth and average degree. In particular, we show that if G has girth at least g and average degree at least Jdn 1 IðgÀ1Þ=2m d; then twðGÞ is at least maxð 2 ; 4eðgþ1Þ ðd À 1Þ À 2Þ: In viewof a well- known conjecture regarding the existence of graphs with girth g; minimum degree d I m and having at most cðd À 1Þ ðgÀ1Þ=2 vertices (for some constant c) (see [7, p. 164]), this lower bound seems to be almost tight (but for a multiplicative factor of g þ 1). In other words, assuming that this conjecture is true, it is easy to construct an infinite family of graphs with girth at least g and average degree at least d but the treewidth I m bounded above by Oððd À 1Þ ðgÀ1Þ=2 Þ (see Section 4 for details). ARTICLE IN PRESS

L.S. Chandran, C.R. Subramanian / Journal of Combinatorial Theory, Series B 93 (2005) 23–32 25

Perhaps, a natural approach to the question of getting a lower bound for the treewidth of G would be to look for the largest r such that G has a Kr minor (Kr stands for the complete graph on r vertices) since the treewidth of a graph is at least the treewidth of any of its minors and twðKrÞ¼r À 1(H is called a minor of G iff H can be obtained from a subgraph of G by a sequence of edge contractions). This problem has received some attention in the literature. It can be inferred from a result 2 ofp Thomassenffiffiffiffiffiffiffiffiffiffi [20] in conjunction with a result in [14,21] that if girthðGÞ is at least cr log r and the minimum degree of G is at least 3 then G has a Kr minor. This result was improved recently by Diestel and Rempel [12] who proved that if girthðGÞ is at least 6 log r þ 3 log log r þ c and dðGÞ is at least 3; then G has a Kr minor. One can infer from the above result that if dðGÞX3; then twðGÞXr0 À 1X2g=6 À 1; where 0 r is the largest integer such that G has a Kr0 minor. Very recently Ku¨ hn and Osthus [15] have improved this result and have shown using probabilistic methods that if girthðGÞ is at least g for some odd g and dðGÞ at least 3; then for some c40; G has a gþ1 cðdffiffiffiffiffiffiffiffiÞ 4 Kr minor for some rXp : Clearly, the lower bound we provide for treewidth in log d terms of girth and minimum degree is much better than (in fact about the square of ) what is derivable from the minor theoretic result of Ku¨ hn and Osthus. More fundamentally, Ku¨ hn and Osthus argue that it is unlikely that their lower bound for the clique minor size may be improved significantly. Thus, extending their point further, it is unlikely that an approach based on clique minors may give a lower bound for the treewidth (in terms of girth and minimum degree or average degree) comparable to ours. For a brief exposition of the early developments on the existence of dense minors in graphs of large girth, see Chapter 8 of [11]. For an introductory account of the role of treewidth in minor theory, see Chapter 12 of [11]. Similar papers which explore the structure of graphs of given girth and average degree (or minimum degree) include [16–18]. The extremal problem of finding upper and lower bounds for the minimum number of vertices that a graph should have so that it has girth g and minimum degree d is also a well-studied one (see [7,10] etc. for a brief history).

2. Preliminaries

Let G be a simple connected undirected graph. Throughout this paper, we will use V for the set of vertices of G and E for the set of edges. jVj will be denoted by n: NðvÞ will denote the set of neighboursS of v i.e., NðvÞ¼fuAV : ðu; vÞAEg: For ADV; we use NðAÞ to denote the set vAA NðvÞÀA: The subgraph of G induced by the vertices in A will be denoted by GðAÞ: pffiffiffiffiffiffiffiffiffiffi 2 The result of [14,21] states that if the average degree of a graph G is at least cr log r then G has a Kr minor. ARTICLE IN PRESS

26 L.S. Chandran, C.R. Subramanian / Journal of Combinatorial Theory, Series B 93 (2005) 23–32

A bijection f : V-f1; 2; y; ng is called an ordering of the vertices of G: Then f ðvÞ is referred to as the number associated with the vertex v; or simply the number of v with respect to the ordering f : Given an ordering f of a graph G; we define the following terms.

Definition 2. Let ADV: The highestðAÞ is defined to be the vertex with the highest number (with respect to f )inA:

Definition 3. A path P ¼ðw1; w2; y; wkÞ in G is called an increasing path, iff f ðw1Þof ðw2Þo?of ðwkÞ: It is called a decreasing path iff f ðw1Þ4f ðw2Þ4?4f ðwkÞ: A single vertex can be considered as either increasing or decreasing.

Definition 4. A vertex uANðvÞ is called a higher neighbour of v iff f ðuÞ4f ðvÞ: The set of higher neighbours of v will be denoted by NhðvÞ; i.e., NhðvÞ¼fuANðvÞ : f ðuÞ4f ðvÞg:

Definition 5. An ordering f of G is called a perfect elimination ordering (PEO) iff for each vAV; the induced subgraph Gðfvg,NhðvÞÞ is a complete subgraph (clique) of G: Then f ðvÞ will be called the PEO number of v:

Note that every graph may not have a PEO. A graph G is called a chordal graph iff every cycle in G of length at least 4 has a chord. (A chord of a cycle C is an edge of G joining two vertices of C which are not consecutive.) The following result is well known (see [13]).

Lemma 1. An undirected graph G is chordal if and only if there exists a PEO for G:

Coming back to treewidth, there are several equivalent characterizations of this concept. Among these, one characterization which is used in this paper is in terms of chordal graphs and is explained below. The clique number oðGÞ is defined to be the number of vertices in a maximum sized clique in G: The following characterization of treewidth is mentioned without proof by Robertson and Seymour [19]. See [11] for a proof.

Lemma 2. For any graph G; twðGÞ is the smallest non-negative integer k; such that G is a subgraph of some chordal graph H with oðHÞ¼k þ 1:

Before proceeding to prove a lower bound for treewidth in terms of girth and average degree, we first give a lower bound in terms of average degree alone. The degeneracy r of a graph G is defined to be the maximum possible minimum degree over all induced subgraphs of G i.e., r ¼ maxH dðHÞ; over all induced subgraphs H of G:

Lemma 3. If r is the degeneracy of a graph G; then twðGÞXr: ARTICLE IN PRESS

L.S. Chandran, C.R. Subramanian / Journal of Combinatorial Theory, Series B 93 (2005) 23–32 27

Proof. Let H be an induced subgraph of G such that the minimum degree of H; dðHÞXr: Then twðGÞXtwðHÞXdðHÞXr; where the first two inequalities follow immediately from the definition of treewidth. &

The following is a simple lemma from extremal graph theory. (See for example, [8, p. 74] or [11].)

Lemma 4. Let d be the average degree of a graph G: Then its degeneracy r is at least Jdn 2 :

XJdn Corollary 1. If G is a graph with average degree d; then twðGÞ 2 :

3. The main lemma

The following lemmata about chordal graphs are useful for us.

Lemma 5 (Chandran [9]). Let P ¼ðw1; w2; y; wkÞ be an induced path in a chordal graph and let wi ¼ highestðPÞ; with respect to a PEO f : Then ðw1; w2; y; wiÞ is an increasing path while ðwi; wiþ1; y; wkÞ is a decreasing path, with respect to f : In other words, the sequence f ðw1Þ; y; f ðwkÞ is unimodal.

Proof. Suppose not. Then there must be some j (where 1ojok) such that f ðwjÀ1Þ4f ðwjÞ and f ðwjþ1Þ4f ðwjÞ: Thus, fwjÀ1; wjþ1gDNhðwjÞ and by the definition of PEO, wjÀ1 must be adjacent to wjþ1; contradicting the assumption that P is an induced path. &

Lemma 6. Let P ¼ðw1; w2; y; wkÞ be an increasing path in a chordal graph with respect to a PEO f : Let f ðuÞ4f ðwkÞ where uANðw1Þ: Then uANhðwkÞ:

Proof. Since w2; uANhðw1Þ; we have uANðw2Þ: The rest follows by induction. &

Lemma 7. Let G ¼ðV; EÞ be a connected chordal graph. Let ADV such that GðAÞ is connected. Let x ¼ highestðAÞ; BDNðAÞ; and z ¼ lowestðBÞ with respect to a PEO f : Then if f ðxÞof ðzÞ; BDNhðxÞ:

Proof. Let yAB: Then there is a wAA such that yANðwÞ: Also f ðyÞXf ðzÞ4f ðxÞ: Consider an induced path P ¼ðw; y; xÞ; which is completely in GðAÞ: Such a path exists, since GðAÞ is connected. Also since x ¼ highestðPÞ; by Lemma 5, P is an increasing path. Then by Lemma 6, yANhðxÞ: Thus we conclude that BDNhðxÞ: &

Definition 6. Let G ¼ðV; EÞ be a simple connected graph on n vertices and s be a number such that 1pspn: We define NminðsÞ¼minjNðXÞj over all (non-empty) X s with 2pjXjps: ARTICLE IN PRESS

28 L.S. Chandran, C.R. Subramanian / Journal of Combinatorial Theory, Series B 93 (2005) 23–32

Lemma 8. Let G ¼ðV; EÞ be a connected chordal graph on n vertices, and let s be an integer such that 1pson: Then, the size of the maximum clique in G; oðGÞXNminðsÞ:

Proof. First note that the lemma is trivial for s ¼ 1: in that case NminðsÞ¼d; the minimum degree of the graph. It is well known that there exists a simplicial vertex3 in every chordal graph and thus the maximum clique size is at least d þ 1: Therefore, we can assume that sX2: Let f be a PEO of G: (A PEO exists for G by Lemma 1.) Let Vi ¼fuAV : f ðuÞpig: Let k be the smallest integer such that GðVkÞ (the induced subgraph on Vk) Xs contains a connected component with number of vertices 2 þ 1: (Note that since sX2; we have kX2:) Let vk be the vertex with PEO number k i.e., f ðvkÞ¼k: Let W1; W2; y; Wt be the subsets of Vk forming the connected components of GðVkÞ and without loss of generality let vk belong to W1: Then clearly W2; W3; y; Wt induce connected components in GðVkÀ1Þ also. Since all the s connected components of GðVkÀ1Þ have less than 2 þ 1 vertices, clearly W1 forms s the connected component of GðVkÞ which has at least 2 þ 1 vertices. Let U1; U2; y; Uh be the subsets of VkÀ1 forming the remaining connected components of GðVkÀ1Þ other than those formed by W2; y; Wt: Then clearly W1 ¼ U1,U2,?,Uh,fvkg: S S D y Xs Let I f1; 2; ; hg beS such that j iAI Uij 2 and j iAI Uij is minimum subject to XsX this condition. Let X ¼ iAI Ui: (X is non-empty since jXj 2 1:) If there exists a j Xs s (where 1pjph) such that jUjj 2; then clearly jXjpjUjjp2 þ 1ps: Otherwise we can s s cA assume that jUjjo2 for all j; 1pjph: Now, since jX À Ucjo2 (for any I)by s X assumption and jUcjo2; we have jXjps again. Therefore jNðXÞj NminðsÞ: Also it is easy to see that NðXÞ-Ui ¼ |; for 1piph; since there cannot be an edge from Ui to Uj since Ui and Uj are assumed to be distinct connected components of GðVkÀ1Þ: We conclude that the PEO number of the lowestðNðXÞÀfvkgÞ4k: Nowconsider the induced subgraph on X,fvkg: Clearly this is connected, and the highestðX,fvkgÞ ¼ vk: Also NðXÞÀfvkgDNðX,fvkgÞ: It follows from Lemma 7 that NðXÞÀfvkgDNhðvkÞ: By the definition of PEO, NhðvkÞ,fvkg induces a clique. Thus NðXÞ induces a clique. Since jNðXÞjXNminðsÞ; we can infer that oðGÞXNminðsÞ: &

Lemma 9. Let G ¼ðV; EÞ be a graph on n vertices and let s be an integer such that 1pspn: Then, twðGÞXNminðsÞÀ1:

Proof. Consider the chordal graph H ¼ðV; E0Þ with the minimum possible maximum clique size (oðHÞ), such that G ¼ðV; EÞ is a subgraph of H: Clearly, 0 since E +E; NminðsÞ with respect to H is at least NminðsÞ with respect to G: Then by Lemma 8, oðHÞXNminðsÞ: Nowthe required result followsfrom Lemma 2. &

3 A vertex v is simplicial if fvg,NðvÞ induces a clique. For example, the vertex with PEO number 1 is simplicial. ARTICLE IN PRESS

L.S. Chandran, C.R. Subramanian / Journal of Combinatorial Theory, Series B 93 (2005) 23–32 29

4. Girth vs treewidth

The following is a recent result from Alon et al. [1]

Lemma 10 (Alon et al. [1]). The number of vertices n in a graph of girth g and average degree at least dX2 satisfies nXNðd; gÞ where

XrÀ1 Nðd; 2r þ 1Þ¼1 þ d ðd À 1Þi; i¼0

XrÀ1 Nðd; 2rÞ¼2 ðd À 1Þi i¼0 for integer rX1:

IgÀ1m Remark. Note that for d42; Nðd; gÞXðd À 1Þ 2 À 2:

Proof of the Remark. When d42; from Lemma 10,

gÀ2 Nðd; gÞX 2ðd À 1Þ 2 À 2 when g is even;

gÀ1 Nðd; gÞX ðd À 1Þ 2 when g is odd:

gÀ2 IgÀ1m & Noting that when g is even, 2 ¼ 2 ; we get the desired result.

Theorem 1. Let G be a graph with girth at least g; average degree at least d: Then d 1 I m twðGÞXmax ; ðd À 1Þ ðgÀ1Þ=2 À 2 2 4eðg þ 1Þ

Proof. Note that in viewof Corollary 1, it is sufficient to consider the case where 1 IðgÀ1Þ=2m Jdn 4eðgþ1Þ ðd À 1Þ À 24 2 : Therefore, without loss of generality we can IgÀ1m assume that d42 þ 2f ; where f ¼ 1 : (If dp2 þ 2f ; then 1 ðd À 1Þ 2 À Igþ1m 4eðgþ1Þ 2 e Jdn 2f X 2p4ðgþ1Þ À 2o1p 2 ; since ð1 þ 2f Þpe ). Moreover, we can assume that g 5: From the definition of treewidth, it is clear that if G0 is an induced subgraph of G; X 0 0 X 1 then twðGÞ twðG Þ: Therefore it is sufficient to prove that twðG Þ 4eðgþ1Þ ðd À I m 1Þ ðgÀ1Þ=2 À 2; for some induced subgraph G0 of G: Let G0 ¼ðV 0; E0Þ be an induced subgraph of G on the minimum possible number of vertices, such that the average degree of G0 is at least d: Clearly girthðG0ÞXgirthðGÞXg (note that since the average degree of G0 is at least dX2; it is not acyclic). ARTICLE IN PRESS

30 L.S. Chandran, C.R. Subramanian / Journal of Combinatorial Theory, Series B 93 (2005) 23–32

0 For any proper subset X of V ; let dX denote the average degree of the induced 0 subgraph on X and let kX be the number of edges (of G ) with exactly one end point in X:

0 Claim 1. Any (non-empty) proper subset X of V satisfies the inequality dX jXjþ 2kX 4djXj:

0 Proof of Claim 1. Suppose dX jXjþ2kX pdjXj forsomepropersubsetX of V : Let 0 0 Y ¼ V À X: Note that the sum of degrees of the vertices in G is equal to dX jXjþ 0 0 0 dY jYjþ2kX : Then clearly, djV jpaverage degree ofG :jV jpdX jXjþ2kX þ 0 dY jYjpdjXjþdY jYj: Thus noting that jV jÀjXj¼jYj; we have djYjpdY jYj; 0 which means dY Xd: Thus Y is a smaller subset of V (compared to V ) such that the average degree of its induced subgraph is at least d; which violates the minimality of G0: IgÀ1m 1 2 Let M ¼ 4e ðd À 1Þ À 2 (note that by the remark following Lemma 10, jV 0j4M). Let f ¼ 1 : & Igþ1m 2

Claim 2. All subsets X of V 0 with jXjpM; satisfy the property jNðXÞjXf jXj:

Proof of Claim 2. Suppose not. Then there exists a subset X of V 0 with jXjpM; but jNðXÞjof jXj: Let H be the induced subgraph on X,NðXÞ: Since dX jXjþ djXj d 2kX 4djXj; the average degree of H is at least jXjþjNðXÞj41þf : Note that since (from d the first paragraph of the proof ) d42ð1 þ f Þ; 1þf 42: Thus, H is not acyclic and its girth is clearly at least g: Applying, Lemma 10 to H; we get Mð1 þ f Þ4 IðgÀ1Þm d 2 jXjþjNðXÞj4ð1þf À 1Þ À 2: Rearranging,

1 IðgÀ1Þm M4 ðd À 1 À f Þ 2 À 2: Igþ1m ð1 þ f Þ 2

Igþ1m 1 Using the inequality ð1 þ xÞpex we have ð1 þ f Þ 2 ¼ð1 þ f Þ f pe: Substituting we get,

IgÀ1m 1 IðgÀ1Þm f 2 M4 ðd À 1Þ 2 1 À À2 e d À 1 dÀ1 1 IgÀ1m f f 4 ðd À 1Þ 2 1 À À2: e d À 1 1 1 À f 4 x 1Àx A X X Note that ð1 À xÞ 4e for x ð0; 1Þ: Take x ¼ dÀ1: Since d 2 þ 2f and g 5 1 f À 1 1 1ÀxX1 (from the first paragraph of this proof ) we have xp1þ2f p2þ1=f p5 we get e 4:

4 y y2 ? x A e ¼ 1 þ y þ 2! þ : Taking y ¼ 1Àx and noting that y40 when x ð0; 1Þ; we can neglect the terms x 1 À1 1Àx 1 x 1Àx beyond the first two and get e 41Àx: Rearranging we get ð1 À xÞ 4e : ARTICLE IN PRESS

L.S. Chandran, C.R. Subramanian / Journal of Combinatorial Theory, Series B 93 (2005) 23–32 31

Thus we have

1 IgÀ1m M4 ðd À 1Þ 2 À 2 4e a contradiction to the assumption. Therefore we conclude that all subsets X of V 0 with jXjpM; satisfy jNðXÞjXf jXj: Nowby Lemma 9, taking s ¼ M; we have

twðGÞX NminðMÞÀ1 M ¼ min jNðXÞj : |aXDV; pjXjpM À 1 2 M X min f jXj : |aXDV; pjXjpM À 1 2 M X f À 1 2 M X À 1 g þ 1 1 I m X ðd À 1Þ ðgÀ1Þ=2 À 2 4eðg þ 1Þ as required. &

Remark on the tightness of the above lower bound. The following is a well-known conjecture on the existence of high-girth graphs (see for example, [7, p. 164]).

Conjecture. There exists a constant c such that for all integers g; dX3; there is a graph Gðg; dÞ of minimum degree at least d and girth at least g whose order (number of IgÀ1m vertices) is at most cðd À 1Þ 2 :

In viewof this conjecture, wesee that the lowerbound given in Theorem 1 is very close to what is best possible (but for a multiplicative factor of g þ 1) since the IgÀ1m treewidth of Gðg; dÞ can be at most cðd À 1Þ 2 ; the total number of vertices, 1 IðgÀ1Þ=2m whereas the lower bound proven by us is at least 4eðgþ1Þ ðd À 1Þ À 2 since the average degree dXd: It may be noted that, by using Gðg; dÞ as components it is easy to construct graphs with minimum degree at least d; girth at least g; and having an arbitrarily large IgÀ1m number of vertices but its treewidth (or pathwidth) still at most cðd À 1Þ 2 : For example let G1; G2; y; Gk be k copies of Gðg; dÞ: Let Ui ¼ðUi0; Ui1; y; UipÞ denote an optimal path decomposition of Gi; and let xi and yi be vertices of Gi such that 0 xiAUi0 and yiAUip: Construct a newgraph G by connecting together G1; G2; y; Gk by adding an edge from yi of Gi to xiþ1 of Giþ1 for i ¼ 0tok À 1: Clearly, U1; S1; U2; S2; y; SkÀ1; Uk where Si ¼fyi; xiþ1g and Ui stands for an optimal ARTICLE IN PRESS

32 L.S. Chandran, C.R. Subramanian / Journal of Combinatorial Theory, Series B 93 (2005) 23–32

0 path decomposition of Gi; is a path decomposition of G : Therefore 0 0 IðgÀ1Þ=2m twðG ÞppathwidthðG ÞppathwidthðGiÞpcðd À 1Þ :

References

[1] N. Alon, S. Hoory, N. Linial, The moore bound for irregular graphs, Graphs Combin. 18 (1) (2002) 53–57. [2] S. Arnborg, Efficient algorithms for combinatorial problems on graphs with bounded decomposa- bility—a survey, BIT 25 (1985) 2–23. [3] S. Arnborg, A. Proskurowski, Linear time algorithms for NP-hard problems on graphs embedded in k-trees, Discrete Appl. Math. 23 (1989) 11–24. [4] H.L. Bodlaender, A tourist guide through treewidth, Acta Cybernet. 11 (1993) 1–21. [5] H.L. Bodlaender, A linear time algorithm for finding tree decompositions of small treewidth, SIAM J. Comput. 25 (1996) 1305–1317. [6] H.L. Bodlaender, T. Hagerup, Parallel algorithms with optimal speedup for bounded treewidth, SIAM J. Comput. 27 (1998) 1725–1746. [7] B. Bollaba´ s, Extremal Graph Theory, Academic Press, NewYork, 1978. [8] J.A. Bondy, Basic graph theory: paths and circuits, in: R. Graham, M. Grotschel, L. Lovasz (Eds.), Handbook of Combinatorics, Vol. 1, Elsevier, Amsterdam, 1995. [9] L.S. Chandran, A linear time algorithm for enumerating all the minimum and minimal separators of a chordal graph, in: Proceedings of the Seventh International Computing and Combinatorics Conference, Lecture Notes in Computer Science, Vol. 2108, Springer, Berlin, 2001, pp. 308–317. [10] L.S. Chandran, A high girth graph construction, SIAM J. Discrete Math. 16 (2003) 366–370. [11] R. Diestel, Graph Theory, Vol. 173, 2nd Edition, Springer, NewYork, 2000. [12] R. Diestel, C. Rempel, Dense minors in graphs of large girth, Combinatorica, to appear. [13] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, NewYork, 1980. [14] A.V. Kostochka, Lower bound of the hadwiger number of graphs by their average degree, Combinatorica 4 (1984) 307–316. [15] D. Ku¨ hn, D. Osthus, Minors in graphs of large girth, Random Structures Algorithms 22 (2003) 213–225. [16] D. Ku¨ hn, D. Osthus, Topological minors in graphs of large girth, J. Combin. Theory B 86 (2002) 364–380. [17] D. Ku¨ hn, D. Osthus, Subdivisions of krþ2 in graphs of average degree at least r þ e and large but constant girth, Combin. Probab. Comput. 13 (2004) 361–371. [18] W. Mader, Topological subgraphs in graphs of large girth, Combinatorica 18 (1998) 405–412. [19] N. Robertson, P.D. Seymour, Graph minors II: algorithmic aspects of tree-width, J. Algorithms 7 (1986) 309–322. [20] C. Thomassen, Girth in graphs, J. Combin. Theory Ser. B 35 (1983) 129–141. [21] A.G. Thomason, An extremal function for contractions of graphs, Math. Proc. Cambridge Philos. Soc. 95 (1984) 261–265.