Degree Type

Thesis

Date of Award

2020

Degree Name

Doctor of Philosophy

Department

Mathematics

Major

Mathematics

First Advisor

Bernard Lidicky

Abstract

In this thesis we discuss two topics: domination parameters and inducibility. In the first chapter, we introduce basic concepts, definitions, and a brief history for both types of problems. We will first inspect domination parameters in graphs, particularly independent domination in regular graphs and we answer a question of Goddard and Henning. Additionally, we provide some constructions for graphs regular graphs of small degree to provide lower bounds on the independent domination ratio of these classes of graphs. In Chapter 3 we expand our exploration of independent domination into the realm of directed graphs. We will prove several results including providing a fastest known algorithm for determining existence of an independent dominating set in directed graphs with minimum in degree at least one and period not eqeual to one. We also construct a set of counterexamples to the analogue of Vizing's Conjecture for this setting. In the fourth chapter, we pivot from independent domination to split domination in directed graphs, where we introduce the split domination sequence. We will determine that almost all possible split domination sequences are realizable by some graphs, and state several open questions that would be of interest to continue on this field. In the fifth chapter we will provide a brief introduction to Flag Algebras, then determine the unique maximizer of induced net graphs in graphs of certain orders.

DOI

https://doi.org/10.31274/etd-20200624-131

Copyright Owner

Adam Blumenthal

Language

en

File Format

application/pdf

File Size

86 pages

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