Regular Matroids with Graphic Cocircuits

In this paper we examine the effect of removing cocircuits from regular matroids and we focus on the case in which such a removal always results in a graphic matroid. The first main result, given in section 3, is that a regular matroid with graphic cocircuits is signed-graphic if and only if it does not contain two specific minors. This provides a useful connection between graphic, regular and signed-graphic matroids which may be further utilised for devising combinatorial recognition algorithms for certain classes of matroids. At this point we should note that decomposition theories and recognition algorithms for matroids have provided some of the most important results of matroid theory and combinatorial optimization (see e.g. decomposition of graphic matroids [16] and recognition of network matrices [2] and decomposition of regular matroids [10] and recognition of totally unimodular matrices [11]). Finally, in section 4 we provide a simple recognition algorithm which determines whether a cographic matroid with graphic cocircuits is signed-graphic or not.