Signed-graphic matroids with all-graphic cocircuits

Abstract

A matroid $M$ has all-graphic cocircuits if $M\setminus Y$ is a graphic matroid for every cocircuit $Y\in {\mathcal{C}}^{\ast }\left(M\right)$. In Papalamprou and Pitsoulis (2013) it has been shown that signed-graphic matroids that are representable in $GF\left(2\right)$ 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.