Dominating Sets of the Cartesian Products of Cycles

Abstract

A dominating set for a graph G is a subset D of V(G) such that every vertex not in D is adjacent to at least one member of D. In this project, we first briefly survey a variety of known results on dominating sets of some families of graphs, especially the Cartesian products of two k-cycles which are our main focus for this project. Then, we describe the application we developed to facilitate research on dominating sets of the Cartesian products of k-cycles. After that, we obtain linear-time algorithms to generate dominating sets of the Cartesian products of two k-cycles with sizes matching the best known upper bounds. Additionally, for two cases when k is congruent to two or three modulo five, we improve the two known upper bounds.