A characterization of graphic matroids using non-separating cocircuits

In this paper, we settle a conjecture made by Wu. We show that a 3-connected binary matroid M is graphic if and only if each element avoids exactly r(M)−1 non-separating cocircuits of M. This result is a natural companion to the following theorem of Bixby and Cunningham: a 3-connected binary matroid M is graphic if and only if each element belongs to exactly 2 non-separating cocircuits of M.