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AbstractBirmele [J Graph Theory 2003] proved that every graph with circumference t has treewidth at most . 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 . Our main result states that, under the additional assumption of ‐connectivity, such graphs have bounded pathwidth. In fact, they have pathwidth . Moreover, examples show that ‐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.
FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO)
FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO)
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