Signed-graphic matroids with all-graphic cocircuits

A matroid M has all-graphic cocircuits if M∖Y is a graphic matroid for every cocircuit Y∈C∗(M). In Papalamprou and Pitsoulis (2013) it has been shown that signed-graphic matroids that are representable in GF(2) can be decomposed into graphic matroids and matroids with all-graphic cocircuits. In this paper we study this class of signed-graphic matroids with all-graphic cocircuits and provide the set of regular excluded minors as well as a complete characterization when the matroid is cographic, which in turn results in a simple recognition algorithm.