On Roundedness in Matroid Theory.

This thesis studies the relationship between subsets and specified minors in a 3-connected matroid. For positive integers k and m, a set S of k-connected matroids is (k,m)-rounded if it satisfies the following condition. Whenever M is a k-connected matroid having an S-minor and X is a subset of E(M) with at most m elements, then M has an S-minor using X. Oxley characterized the (3,2)-rounded sets that contain a single matroid. In Chapter 2, we obtain an analog of this result for binary matroids. In Chapter 3, we use this result to characterize the pairs of matroids which form (3,2)-rounded sets. The methods of Chapter 3 are generalized to 4-connected matroids in Chapter 4 to determine the (4,2)-rounded sets that contain a single matroid. This extends results of Coullard and Kahn. For a 3-connected minor N of a 3-connected matroid M, the following question arises from roundedness theory. Let X be a subset of E(M). How small a 3-connected minor of M can we find which both uses X and has an N-minor? Seymour answered this question for $\vert$X$\vert$ = 1 and 2. We answer this question for $\vert$X$\vert$ $\geq$ 3 in Chapter 5. Finally, in Chapter 6, results from roundedness theory are applied to the study of 3-element circuits in 3-connected matroids. An extension of a result of Asano, Nishizeki, and Seymour is obtained for binary matroids which are non-regular