Date of Award

6-2021

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Dr. Ping Zhang

Second Advisor

Dr. John Martino

Third Advisor

Dr. Gary Chartrand

Fourth Advisor

Dr. Dinesh Sarvate

Keywords

Dominating functions, proper dominating functions, regular dominating functions, total dominating functions, antiregular dominating functions, dominating functions of trees

Abstract

Domination in graphs has become one of the most popular areas of graph the- ory, no doubt due to its many fascinating problems and applications to modern society, as well as the sheer mathematical beauty of the subject. While this area evidently began with the work by the French mathematician Claude Berge in 1958 and the Norwegian-American mathematician Oystein Ore in 1962, domination did not become an active area of research until 1977 with the appearance of the survey paper by Ernest Cockayne and Stephen Hedetniemi. Since then a large number of variations of domination have surfaced and provided numerous applications to dif- ferent areas of science and real-life problems. In 1987, Hedetniemi introduced the concept of dominating functions which (a) provided an analytic method of study- ing this discrete concept, (b) built a connection between domination and graph labelings and colorings, and (c) gave rise to new dominating function parameters. In 2019, Gary Chartrand introduced several variations of dominating functions in connection with some of the best-known concepts in graphs, including irregular- ity, regularity and chromaticity in graphs. In this research, we study irregular, antiregular, regular and proper dominating functions as well as total dominating functions in graphs. We present structural and extremal results dealing with these domination parameters and establish relationships between the concepts of various dominating functions and some well-known parameters in graphs.

Access Setting

Dissertation-Open Access

Download

Share

COinS