### Abstract:

In this thesis we will present two main theorems that can be used to study
minor minimal non even cut matroids.
Given any signed graph we can associate an even cut matroid. However, given
an even cut matroid, there are in general, several signed graphs which
represent that matroid. This is in contrast to, for instance graphic (or
cographic) matroids, where all graphs corresponding to a particular
graphic matroid are essentially equivalent. To tackle the multiple
non equivalent representations of even cut matroids we use the concept of
Stabilizer first introduced by Wittle. Namely, we show the following:
given a "substantial" signed graph, which represents a matroid N that is a
minor of a matroid M, then if the signed graph extends to a signed graph
which represents M then it does so uniquely. Thus the representations of the
small matroid determine the representations of the larger matroid containing
it. This allows us to consider each representation of an even cut matroid
essentially independently.
Consider a small even cut matroid N that is a minor of a matroid M that is
not an even cut matroid. We would like to prove that there exists a
matroid N' which contains N and is contained in M such that the size of N'
is small and such that N' is not an even cut matroid (this would imply in
particular that there are only finitely many minimally non even cut
matroids containing N). Clearly, none of the representations of N extends to
M. We will show that (under certain technical conditions) starting from a
fixed representation of N, there exists a matroid N' which contains N
and is contained in M such that the size of N' is small and such that the
representation of N does not extend to N'.