Removable Circuits in Binary Matroids

We show that if M is a connected binary matroid of cogirth at least five which does not have both an F 7 -minor and an F 7 -minor, then M has a circuit C such that M \Gamma C is connected and r(M \Gamma C) = r(M ). 1 Introduction We shall consider the problem of finding sufficient conditions for the existence of a circuit in a given matroid M whose deletion leaves the rank or connectivity of M unchanged. The existence of such a circuit in graphs has been considered by various authors. The most general result for simple graphs can be deduced from a theorem of W. Mader [5, Satz 1]. Theorem 1 Let k be a positive integer and G be a simple k-connected graph of minimum degree at least k+2. Then G has a circuit C such that G \Gamma E(C) is k-connected. Stronger results for the special case when G is simple and k = 2 can be found in Jackson [4] and Thommassen and Toft [10]. It seems natural to ask if Theorem 1 can be extended to a graph G, which may contain multiple edges. We can obt..