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 - unrestricted
Recommended Citation
Jackson, Eugenie Marie, "Explorations in the Classification of Vertices as Good or Bad." (2001). Electronic Theses and Dissertations. Paper 132. https://dc.etsu.edu/etd/132
Copyright
Copyright by the authors.