Applications and Variations of Domination in Graphs

OF THE DISSERTATION Applications and Variations of Domination in Graphs by Paul Andrew Dreyer, Jr. Dissertation Director: Fred S. Roberts In a graph G = (V, E), S ⊆ V is a dominating set of G if every vertex is either in S or joined by an edge to some vertex in S. Many different types of domination have been researched extensively. This dissertation explores some new variations and applications of dominating sets. We first introduce the concept of Roman domination. A Roman dominating function is a function f : V → {0, 1, 2} such that every vertex v for which f(v) = 0 has a neighbor w with f(w) = 2. This corresponds to a problem in army placement where every region is either defended by its own army or has a neighbor with two armies, in which case one of the two armies can be sent to the undefended region if a conflict breaks out. The weight of a Roman dominating function f is f(V ) = ∑ v∈V f(v), and we are interested in finding Roman dominating functions of minimum weight. We explore the graph theoretic, algorithmic, and complexity issues of Roman domination, including algorithms for finding minimum weight Roman dominating functions for trees and grids. We then explore a dynamic variant of domination. Given a graph with each vertex having a sign {+,−}, we define a k−threshold process by updating the graph at every time step according to the rule that a vertex switches sign if and only if k or more of its neighbors have the opposite sign. A set S of vertices is a k−conversion set if when a