Isomorphism for even cycle matroids - I

TL;DR

This paper investigates the isomorphism problem for even cycle matroids, providing solutions for two classes of signed graphs and proposing a conjecture about the general structure of such representations.

Contribution

It introduces solutions for the isomorphism problem within two specific classes of signed graphs representing even cycle matroids and conjectures a broader classification.

Findings

Solved the isomorphism problem for two classes of signed graphs

Proposed a conjecture on the structure of all signed graph representations

Identified operations relating different signed graph representations

Abstract

A seminal result by Whitney describes when two graphs have the same cycles. We consider the analogous problem for even cycle matroids. A representation of an even cycle matroid is a pair formed by a graph together with a special set of edges of the graph. Such a pair is called a signed graph. We consider the problem of determining the relation between two signed graphs representing the same even cycle matroid. We refer to this problem as the Isomorphism Problem for even cycle matroids. We present two classes of signed graphs and we solve the Isomorphism Problem for these two classes. We conjecture that, up to simple operations, any two signed graphs representing the same even cycle matroid are either in one of these classes, or related by a modification of an operation for graphic matroids, or belonging to a small set of examples.