LSU Historical Dissertations and Theses

1995

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Mathematics

James G. Oxley

Abstract

This dissertation solves some problems related to the structure of matroids. In Chapter 2, we prove that if M and N are distinct connected matroids on a common ground set E, where \$\vert E\vert \ge 2,\$ and, for every e in \$E,\ M\\ e = N\\ e\$ or M/e = N/e, then one of M and N is a relaxation of the other. In addition, we determine the matroids M and N on a common ground set E such that, for every pair of elements \$\{ e,f\}\$ of E, at least two of the four corresponding minors of M and N obtained by eliminating e and f are equal. The theorems in Chapter 3 and 4 extend a result of Oxley that characterizes the non-binary matroids M such that, for each element e, \$M\\ e\$ or M/e is binary. In Chapter 3, we describe the non-binary matroids M such that, for every pair of elements \$\{ e,f\} .\$ at least two of the four minors of M obtained by eliminating e and f are binary. In Chapter 4, we obtain an alternative extension of Oxley's result by changing the minor under consideration from the smallest 3-connected whirl, \$U\sb{2,4},\$ to the smallest 3-connected wheel, \$M(K\sb4).\$ In particular, we determine the binary matroids M having an \$M(K\sb4)\$-minor such that, for every element e, \$M\\ e\$ or M/e has no \$M(K\sb4)\$-minor. This enables us to characterize the matroids M that are not series-parallel networks, but, for every \$e,\ M\\ e\$ or M/e is a series-parallel network.

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