Circumference and Pathwidth of Highly Connected Graphs

Abstract

Birmele [J. Graph Theory, 2003] proved that every graph with circumference t has treewidth at most t − 1. Under the additional assumption of 2-connectivity, such graphs have bounded pathwidth, which is a qualitatively stronger conclusion. Birmele's theorem was extended by Birmele, Bondy and Reed [Combinatorica, 2007] who showed that every graph without k disjoint cycles of length at least t has treewidth O(tk 2 ). Our main result states that, under the additional assumption of (k + 1)-connectivity, such graphs have bounded pathwidth. In fact, they have pathwidth O(t 3 +tk 2 ). Moreover, examples show that (k+1)-connectivity is required for bounded pathwidth to hold. These results suggest the following general question: for which values of k and graphs H does every k-connected H-minor-free graph have bounded pathwidth? We discuss this question and provide a few observations.

Long Cycles have the Edge-Erdős-Pósa Property

“…In addition, F ′ ∩ ε(T w ) is an ε(w)-linked B ν -tree. This proves (7). Now let F be an ε(v)-linked B µ -tree in ε(T v ).…”

“…Robertson and Seymour [11] were the first to prove that a graph of large pathwidth contains a subdivision of a binary tree with large height. Marshall and Wood [7] prove an explicit formula if the graph is a tree: Lemma 11 (Marshall and Wood [7], restated). Let T be a tree with at least two vertices.…”

“…Then it lies in ε(T w ) for some ≤ T -maximal vertex in V (T v − v) ∩ U . Then, by (7), the graph ε(T w ) contains an ε(w)-linked B µ -tree. In this case, we replace v by w and proceed with the proof.…”

“…Observe that Lemma 14 yields two distinct components K 1 , K 2 of G − ε(v) and vertices x 1 , x 2 such that ε(T wi ) lies in K i and x i separates every vertex of degree at least 3 of K i from G − K i . Now, for each of i ∈ [2], we can apply (7) with ν = µ − 1 ≥ 2 in order to find an ε(w i )-linked B µ−1 -tree in ε(T wi ). With induction on µ we then find for each i ∈ [2] a w i -linked B µ−1 -tree in T wi , which finishes the proof.…”

Directed width parameters and circumference of digraphs

“…Nesetril and Ossona de Mendez [NdM12] showed that the pathwidth of a 2-connected graph G is at most (circ(G) − 2) 2 . Marshall and Wood [MW13] improved this bound to circ(G)/2 (circ(G)−1). Generalizing these results to directed pathwidth, under a suitable directed connectivity assumption is an interesting open problem.…”