## 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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