Stabilizer theorems for even cycle matroids

Abstract

A signed graph is a representation of an even cycle matroid M if the cycles of M correspond to the even cycles of that signed graph. Two long standing open questions regarding even cycle 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 cycle matroids can have an arbitrary number of pairwise inequivalent representations (for a natural definition of equivalence). We show that we can bound the number of inequivalent representations of an even cycle matroid M (under some mild connectivity assumptions) if M contains any fixed size minor that is not a projection of a graphic matroid. For instance, any connected even cycle matroid which contains R10 as a minor has at most 6 inequivalent representations.