Asteroidal Sets and Dominating Targets in Graphs

Authors

Oleksiy Al-saadi, University of Nebraska-LincolnFollow

First Advisor

Andrew J. Radcliffe

Date of this Version

5-2024

Document Type

Article

Citation

A dissertation presented to the faculty of the Graduate College at the University of Nebraska in partial fulfillment of requirements for the degree of Doctor of Philosophy

Major: Computer Science

Under the supervision of Professor Andrew J. Radcliffe

Lincoln, Nebraska, May 2024

Comments

Copyright 2024, Oleksiy Al-saadi. Used by permission

Abstract

The focus of this PhD thesis is on various distance and domination properties in graphs. In particular, we prove strong results about the interactions between asteroidal sets and dominating targets. Our results add to or extend a plethora of results on these properties within the literature. We define the class of strict dominating pair graphs and show structural and algorithmic properties of this class. Notably, we prove that such graphs have diameter 3, 4, or contain an asteroidal quadruple. Then, we design an algorithm to to efficiently recognize chordal hereditary dominating pair graphs. We provide new results that describe the internal structure of these graphs, and prove that asteroidal quadruples may provide diameter bounds. Then, we extend the notion of polarity to dominating targets by defining the concept of polar targets. We investigate dominating targets in cycle graphs and show that they cannot have polar targets. Then, we provide a sufficient condition for a graph to have a polar target of size 3.

Advisor: Andrew J. Radcliffe