On cocircuit covers of bicircular matroids

Abstract

Given a graph $G$, one can define a matroid $M=(E,\mathcal{C})$ on the edges $E$ of $G$ with circuits $\mathcal{C}$ where $\mathcal{C}$ is either the cycles of $G$ or the bicycles of $G$. The former is called the cycle matroid of $G$ and the latter the bicircular matroid of $G$. For each bicircular matroid $B(G)$, we find a cocircuit cover of size at most the circumference of $B(G)$ that contains every edge at least twice. This extends the result of Neumann-Lara, Rivera-Campo and Urrutia for graphic matroids.