Pathwidth versus cocircumference

The circumference of a graph G with at least one cycle is the length of a longest cycle in G. A classic result of Birmele'\ [J. Graph Theory, 43 (2003), pp. 24--25] states that the treewidth of G is at most its circumference minus 1. In case G is 2 -connected, this upper bound also holds for the pathwidth of G; in fact, even the treedepth of G is upper bounded by its circumference (Brianski 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 G by its cocircumference, defined as the largest size of a bond in G, an inclusionwise minimal set of edges F such that G - F has more components than G. In matroidal terms, the cocircumference of G is the circumference of the bond matroid of G. Our first result is the following ``dual"" version of Birmele'\'s theorem: The treewidth of a graph G is at most its cocircumference. Our second and main result is an upper bound of 3k - 2 on the pathwidth of a 2 -connected graph G with cocircumference k. Contrary to circumference, no such bound holds for the treedepth of G. Our two upper bounds are best possible up to a constant factor.