Relationships between Pairs of Representations of Signed Binary Matroids

Abstract

We show how pairs of signed graphs with the same even cycles relate to pairs of grafts with the same even cuts. These results are proved in the more general context of signed binary matroids

Stabilizer theorems for even cycle matroids

“…The following easy observation is the analogue to Remark 17 for the case of even cut matroids (see [4]). …”

“…We believe that there exists a constant k such that every even cycle matroid with more than k inequivalent representations is constructed in a way analogous to that of the example in Section 1.2. The problem of describing the pairwise relationship between any two representations of an even cycle matroid is discussed in [4].…”

“…It is easy to show that Lovász-flips preserve even cycles [3,4]. Using Lovász-flips we can construct inequivalent signed graphs representing the same even cycle matroid.…”

“…The odd edges, in both signed graphs, are 1, 2, 3. Note that, for every i ∈ [4], the two vertices in V G i ∩V G i+1 form a blocking pair. It is possible to obtain the signed graph on the right from the signed graph on the left by signature exchanges and Lovász-flips on each of these blocking pairs.…”

Displaying blocking pairs in signed graphs

“…Unfortunately, there is no simple description of the set of all representations of an even cycle matroid [8,13]. Suppose that (G 1 , ⌃ 1 ) and (G 2 , ⌃ 2 ) are signed graphs where G 1 and G 2 are equivalent and ⌃ 2 is obtained from ⌃ 1 by resigning.…”