Structural Results for Matroids.

Abstract

This dissertation solves some problems involving the structure of matroids. In Chapter 2, the class of binary matroids with no minors isomorphic to the prism graph, its dual, and the binary affine cube is completely determined. This class contains the infinite family of matroids obtained by sticking together a wheel and the Fano matroid across a triangle, and then deleting an edge of the triangle. In Chapter 3, we extend a graph result by D. W. Hall to matroids. Hall proved that if a simple, 3-connected graph has a $K\sb5$-minor, then it must also have a $K\sb{3,3}$-minor, the only exception being $K\sb5$ itself. We prove that if a 3-connected, binary matroid has an $M(K\sb5)$-minor, then it must also have a minor isomorphic to $M(K\sb{3,3})$ or its dual, the only exceptions being $M(K\sb5),$ a highly symmetric 12-element matroid $T\sb{12},$ and $T\sb{12}$ with any edge contracted. Chapter 4 consists of a collection of results on the intersection of circuits and cocircuits in binary matroids. In Chapter 5, we describe, in terms of excluded minors, the class of non-binary matroids with the property that a matroid is in the class if its restriction to every hyperplane is binary.