  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.
