Global ETD Search

1	Explorations in the Classification of Vertices as Good or Bad. Jackson, Eugenie Marie 01 May 2001 (has links) (PDF) 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. graph theory bad vertices domination domination-fair graphs realizability caterpillars paths domination-commendable graphs domination-excellent graphs goodness index good vertices domination-poor graphs Physical Sciences and Mathematics