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