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 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 R 10 as a minor has at most 6 inequivalent representations.

Displaying blocking pairs in signed graphs

“…We say that (G, ⌃) extends the representation (H, ) of N to the major M . The following is proved in [9].…”

“…An even cycle is non-degenerate if none of its representation has a blocking pair. The following was proved in [9]. Theorem 9.…”

Recognizing Even-Cycle and Even-Cut Matroids

“…We will call such a pair of signed-graphs, equivalent. Note that this is indeed an equivalence relation (see [10]), thus we can partition the set of all representations of an evencycle matroid into equivalence classes.…”

“…) and (G , T ) are equivalent then ecut(G, T ) = ecut(G , T ) since cocycles of ecut(G, T ) are precisely the cycles of G and the T -joins of (G, T ). Note that this is indeed an equivalence class (see [6]), thus we can partition the set of all representations of an even-cut matroid into equivalence classes. Now, we will introduce an example with multiple inequivalent representations.…”

“…We illustrate this shuffling in Figure 1.5. It can be readily checked in [10] that ecycle(G, Σ) = ecycle(G , Σ ). It is now straightforward to see that we can have an exponential number (in the number of elements) of inequivalent signed-graph representations all related by shuffling subsets of edges.…”