Pathwidth Versus Cocircumference

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 857-866, March 2024.
Abstract. The circumference of a graph [math] with at least one cycle is the length of a longest cycle in [math]. A classic result of Birmelé [J. Graph Theory, 43 (2003), pp. 24–25] states that the treewidth of [math] is at most its circumference minus 1. In case [math] is 2-connected, this upper bound also holds for the pathwidth of [math]; in fact, even the treedepth of [math] is upper bounded by its circumference (Briański et al. [Treedepth vs circumference, Combinatorica, 43 (2023), pp. 659–664]). In this paper, we study whether similar bounds hold when replacing the circumference of [math] by its cocircumference, defined as the largest size of a bond in [math], an inclusionwise minimal set of edges [math] such that [math] has more components than [math]. In matroidal terms, the cocircumference of [math] is the circumference of the bond matroid of [math]. Our first result is the following “dual” version of Birmelé’s theorem: The treewidth of a graph [math] is at most its cocircumference. Our second and main result is an upper bound of [math] on the pathwidth of a 2-connected graph [math] with cocircumference [math]. Contrary to circumference, no such bound holds for the treedepth of [math]. Our two upper bounds are best possible up to a constant factor.