Stabilizer theorems for even cycle matroids

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 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 (two signed graph are equivalent if they are related by a sequence of Whitney-flips and signature exchanges). 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. ∗Present address: Dept. of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada; email:[email protected]; phone:(+1)778 782 5754; fax:(+1)778 782 3332.