Treedepth vs circumference

11/21/2022

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by   Marcin Briański, et al.

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The circumference of a graph G is the length of a longest cycle in G, or +∞ if G has no cycle. Birmelé (2003) showed that the treewidth of a graph G is at most its circumference minus 1. We strengthen this result for 2-connected graphs as follows: If G is 2-connected, then its treedepth is at most its circumference. The bound is best possible and improves on an earlier quadratic upper bound due to Marshall and Wood (2015).