#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### Abstract

This dissertation is a collection of work on matroid and polymatroid connectivity. Connectivity is a useful property of matroids that allows a matroid to be decomposed naturally into its connected components, which are like blocks in a graph. The Cunningham-Edmonds tree decomposition further gives a way to decompose matroids into 3-connected minors. Much of the research below concerns alternate senses in which matroids and polymatroids can be connected. After a brief introduction to matroid theory in Chapter 1, the main results of this dissertation are given in Chapters 2 and 3. Tutte proved that, for an element e of a 2- connected matroid M , either the deletion or the contraction of e for M is 2-connected. In Chapter 2, a new notion of matroid connectivity is defined and it is shown that this new notion only enjoys the above inductive property when it agrees with the usual notion of 2-connectivity. Another result is proved to reinforce the special importance of this usual notion. In Chapter 3, a result of Brylawski and Seymour is considered. That result extends Tutte’s theorem by showing that if the element e is chosen to avoid a 2-connected minor N of M, then the deletion or contraction of e form M is not only 2-connected but maintains N as a minor. The main result of Chapter 3 proves an analogue of this result for 2-polymatroids, a natural extension of matroids. Chapter 4 describes a class of binary matroids that generalizes cubic graphs. Specifically, attention is focused on binary matroids having a cocircuit basis where every cocircuit in the basis, as well as the symmetric difference of all these cocircuits, has precisely three elements.

#### Date

3-17-2021

#### Recommended Citation

Gershkoff, Zachary R., "Connectivity of Matroids and Polymatroids" (2021). *LSU Doctoral Dissertations*. 5494.

https://digitalcommons.lsu.edu/gradschool_dissertations/5494

#### Committee Chair

Oxley, James