Enumerating matroid extensions

Abstract

This thesis investigates the problem of enumerating the extensions of certain matroids. A matroid M is an extension of a matroid N if M delete e is equal to N for some element e of M. Similarly, a matroid M is a coextension of a matroid N if M contract e is equal to N for some element e of M. In this thesis, we consider extensions and coextensions of matroids in the classes of graphic matroids, representable matroids, and frame matroids. We develop a general strategy for counting the extensions of matroids which translates the problem into counting stable sets in an auxiliary graph. We apply this strategy to obtain asymptotic results on the number of extensions and coextensions of certain graphic matroids, projective geometries, and Dowling geometries.