Pathwidth vs cocircumference
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The circumference of a graph G with at least one cycle is the length of a longest cycle in G. A classic result of Birmelé (2003) 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 (Briański, Joret, Majewski, Micek, Seweryn, Sharma; 2023). 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 inclusion-wise 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 Birmelé'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.
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