Circumference and Pathwidth of Highly Connected Graphs

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 et al. [Combinatorica 2007] who showed that every graph without k disjoint cycles of length at least t has treewidth O(tk2). 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 + tk2). Moreover, examples show that (k + 1)-connectivity is required for bounded pathwidth to hold. These results suggest the following general question: for which Contract grant sponsor: National Science Foundation; contract grant number: 1310758; Contract grant number: Australian Research Council. Journal of Graph Theory C © 2014 Wiley Periodicals, Inc.