#### Date of Award

4-1-2023

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Ping Zhang, Ph.D.

#### Second Advisor

Gary Chartrand, Ph.D.

#### Third Advisor

Clifton Ealy, Ph.D.

#### Fourth Advisor

Dinesh Sarvate, Ph.D.

#### Keywords

Domination, graph theory, irregular domination

#### Abstract

Domination in graphs has been a popular area of study due in large degree to its applications to modern society as well as the mathematical beauty of the topic. While this area evidently began with the work of Claude Berge in 1958 and Oystein Ore in 1962, domination did not become an active area of research until 1977 with the appearance of a survey paper by Ernest Cockayne and Stephen Hedetniemi. Since then, a large number of variations of domination have surfaced and provided numerous applications to different areas of science and real-life problems. Among these variations are domination parameters defined in terms of distance which provide a more general setting for domination in graphs.

In this research, we study a variant based around distance of graph domination based around distance referred to as irregular domination. A set S of vertices in a connected graph G is an irregular domination set if the vertices of S can be labeled with distinct positive integers in such a way that for every vertex u of G, there is a vertex v ∈ S such that the distance from u to v is the label assigned to v. The minimum cardinality of an irregular dominating set in a graph G is the irregular domination number of G. If for every vertex v ∈ S, there is a vertex u of G such that v is the only vertex of S whose distance to u is the label of v, then S is a minimal irregular dominating set. A graph H is an irregular domination graph if there exists a graph G with a minimal irregular dominating set S such that H is isomorphic to the subgraph of G induced by S. We investigate the irregular domination number of some well-known classes of bipartite graphs and determine all irregular domination paths, cycles, and the familiar graphs of ladders and prisms. Furthermore, we establish characterizations of irregular domination trees, forests, and disconnected graphs. Additionally, we explore connections between irregular dominating sets and irregular domination graphs; other structural results and problems dealing with irregular domination are presented.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Mays, Caryn, "Irregular Domination in Graphs" (2023). *Dissertations*. 3945.

https://scholarworks.wmich.edu/dissertations/3945