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,à 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, ÃCorresponding 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.