Degree Name

MS (Master of Science)

Program

Mathematical Sciences

Date of Award

5-2001

Committee Chair or Co-Chairs

Teresa W. Haynes

Committee Members

Anant P. Godbole, Debra J. Knisley, Robert B. Gardner

Abstract

For a graph G, a set S is a dominating set if every vertex in V-S has a neighbor in S. A vertex contained in some minimum dominating set is called good; otherwise it is bad. A graph G has g(G) good vertices and b(G) bad vertices. The relationship between the order of G and g(G) assigns the graph to one of four classes.

Our results include a method of classifying caterpillars. Further, we develop realizability conditions for a graph G given a triple of nonnegative integers representing the domination number of γ(G), g(G), and b(G), respectively, and provide constructions of graphs meeting those conditions. We define the goodness index of a vertex v in a graph G as the ratio of distinct γ(G)-sets containing v to the total number of γ(G)-sets, and provide formulas that yield the goodness index of any vertex in a given path.

Document Type

Thesis - Open Access

Copyright

Copyright by the authors.

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