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Mathematics, Combinatorics, math.CO,Computer Science, Discrete Mathematics, cs.DM
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 one. 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).
