  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.
