Stabilizer theorems for even cut matroids

A graft is a representation of an even cut matroid M if the cycles of M correspond to the even cuts of the graft. Two, long standing, open questions regarding even cut matroids are the problem of finding an excluded minor characterization and the problem of efficiently recognizing this class of matroids. Progress on these problems has been hampered by the fact that even cut matroids can have an arbitrary number of pairwise inequivalent representations (two grafts are equivalent if the underlying graphs are related by Whitney-flips and the grafts have the same T -joins). We show that we can bound the number of inequivalent representations of an even cut matroid M (under some connectivity assumptions) if M contains any fixed size minor that is not a projection of a co-graphic matroid. For instance, any connected even cut matroid which contains R10 as a minor has at most 10 inequivalent representations.