Excluded-Minors for The Class of Graphic Splitting Matroids

Abstract

This paper is based on the splitting operation for binary matroids that was introduced by Raghunathan, Shikare and Waphare [ Discrete Math. \(184 (1998)\), p.\(267-271\)] as a natural generalization of the corresponding operation in graphs. Here, we consider the problem of determining precisely which graphs \(G\) have the property that the splitting operation, by every pair of edges, on the cycle matroid \(M(G)\) yields a graphic matroid. This problem is solved by proving that there are exactly four minor-minimal graphs that do not have this property.