Parity Domination in Product Graphs

Abstract

iv 1 Preliminaries 1 2 Odd Open Dominating Sets in the Direct Product of Graphs 4 2.1 Odd Open Dominating Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Direct Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Odd Closed r-Dominating Sets in Strong Products of Graphs 11 3.1 Odd Closed r-Dominating Sets . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 The Strong Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 The Problem of Enumeration 20 Bibliography 24 Vita 26 Abstract PARITY DOMINATION IN PRODUCT GRAPHS By Christopher Alan Whisenant, Master of Science. A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University. Virginia Commonwealth University, 2011. Director: Dewey T. Taylor, Associate Professor, Department of Mathematics and Applied Mathematics.PARITY DOMINATION IN PRODUCT GRAPHS By Christopher Alan Whisenant, Master of Science. A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University. Virginia Commonwealth University, 2011. Director: Dewey T. Taylor, Associate Professor, Department of Mathematics and Applied Mathematics. An odd open dominating set of a graph is a subset of the graph’s vertices with the property that the open neighborhood of each vertex in the graph contains an odd number of vertices in the subset. An odd closed r-dominating set is a subset of the graph’s vertices with the property that the closed r-ball centered at each vertex in the graph contains an odd number of vertices in the subset. We first prove that the n-fold direct product of simple graphs has an odd open dominating set if and only if each factor has an odd open dominating set. Secondly, we prove that the n-fold strong product of simple graphs has an odd closed r-dominating set if and only if each factor has an odd closed r-dominating set.