The {\em circumference} of a graph $G$ with at least one cycle is the length of a longest cycle in $G$. A classic result of Birmel\'e (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 (Bria\'nski, Joret, Majewski, Micek, Seweryn, Sharma; 2023). In this paper, we study whether similar bounds hold when replacing the circumference of $G$ by its {\em cocircumference}, defined as the largest size of a {\em 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 Birmel\'e'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.

翻译：具有至少一个环路的图$G$的{\em周长}定义为$G$中最长环路的长度。Birmelé（2003）的一个经典结论指出，$G$的树宽最多为其周长减$1$。若$G$是$2$-连通的，此上界同样适用于$G$的路径宽度；事实上，$G$的树深也受其周长的上界约束（Briański, Joret, Majewski, Micek, Seweryn, Sharma; 2023）。本文研究当将$G$的周长替换为{\em共周长}（定义为$G$中最大{\em键}的规模，即边集$F$的包含关系下极小集合，使得$G-F$的连通分量多于$G$）时，是否仍存在类似上界。在拟阵术语中，$G$的共周长等价于$G$的键拟阵的周长。我们的第一个结果是Birmelé定理的以下"对偶"版本：图$G$的树宽最多为其共周长。第二个且主要的结果是：对于共周长为$k$的$2$-连通图$G$，其路径宽度存在上界$3k-2$。与周长不同，$G$的树深并无此类上界。这两个上界在常数因子意义下是最优的。