#### Date of Award

1995

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

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.

#### Recommended Citation

Mills, Allan Donald, "The Determination of a Matroid's Structure From Properties of Certain Large Minors." (1995). *LSU Historical Dissertations and Theses*. 6036.

https://digitalcommons.lsu.edu/gradschool_disstheses/6036

#### Pages

96