Matroid Connectivity.

Abstract

This dissertation has three parts. The first part, Chapter 1, considers the coefficient $b\sb{ij}(M)$ of $x\sp{i}y\sp{j}$ in the Tutte polynomial of a connected matroid M. The main result characterizes, for each i and j, the minor-minimal such matroids for which $b\sb{ij}(M)>0.$ One consequence of this characterization is that $b\sb{11}(M)>0$ if and only if the two-wheel is a minor of M. Similar results are obtained for other values of i and j. These results imply that if M is simple and representable over $GF(q),$ then there are coefficients of its Tutte polynomial which count the flats of M that are projective spaces of specified rank. Similarly, for a simple graphic matroid $M(G),$ there are coefficients that count the number of cliques of each size contained in G. The second part, Chapter 2, generalizes a graph result of Mader by proving that if f is an element of a circuit C of a 3-connected matroid M and $M\\ e$ is not 3-connected for each $e\in C-\{f\},$ then C meets a triad of M. Several consequences of this result are proved. One of these generalizes a graph result of Wu. The others provide 3-connected analogues of 2-connected matroid results of Oxley. The third part, Chapters 3-5, involves a decomposition of 3-connected binary matroids based on 3-separations and three-sums. The dual of this decomposition is a direct generalization of a decomposition due to Coullard, Gardner, and Wagner for 3-connected graphs. In Chapter 3, we define the decomposition and prove that minimal such decompositions are unique. In Chapter 4, the components of this decomposition are characterized. In Chapter 5, it is shown that, when restricted to contraction-minimally 3-connected binary matroids, the components that are not vertically 4-connected are wheels, duals of twirls, or binary spikes.