In this thesis we will present two main theorems that can be used to studyminor minimal non even cut matroids. Given any signed graph we can associate an even cut matroid. However, givenan even cut matroid, there are in general, several signed graphs whichrepresent that matroid. This is in contrast to, for instance graphic (orcographic) matroids, where all graphs corresponding to a particulargraphic matroid are essentially equivalent. To tackle the multiplenon equivalent representations of even cut matroids we use the concept ofStabilizer first introduced by Wittle. Namely, we show the following:given a ;;substantial;; signed graph, which represents a matroid N that is aminor of a matroid M, then if the signed graph extends to a signed graphwhich represents M then it does so uniquely. Thus the representations of thesmall matroid determine the representations of the larger matroid containingit. This allows us to consider each representation of an even cut matroidessentially independently.Consider a small even cut matroid N that is a minor of a matroid M that isnot an even cut matroid. We would like to prove that there exists amatroid 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 inparticular that there are only finitely many minimally non even cutmatroids containing N). Clearly, none of the representations of N extends toM. We will show that (under certain technical conditions) starting from afixed representation of N, there exists a matroid N;; which contains Nand is contained in M such that the size of N;; is small and such that therepresentation of N does not extend to N;;.
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