Treedepth vs circumference

TL;DR

This paper establishes a tight upper bound on the treedepth of 2-connected graphs based on their circumference, improving previous quadratic bounds and deepening understanding of graph parameters.

Contribution

It proves that for 2-connected graphs, treedepth is at most the circumference, strengthening prior results and providing the best possible bound.

Findings

Treedepth of 2-connected graphs is at most their circumference.

The bound is tight and cannot be improved.

Improves previous quadratic bounds by Marshall and Wood.

Abstract

The circumference of a graph $G$ is the length of a longest cycle in $G$, or $+\infty$ if $G$ has no cycle. Birmel\'e (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).