Power domination in cylinders, tori, and generalized Petersen graphs

Abstract

AbstractA set S of vertices is defined to be a power dominating set (PDS) of a graph G if every vertex and every edge in G can be monitored by the set S according to a set of rules for power system monitoring. The minimum cardinality of a PDS of G is its power domination number. In this article, we find upper bounds for the power domination number of some families of Cartesian products of graphs: the cylinders Pn□Cm for integers n ≥ 2, m ≥ 3, and the tori Cn□Cm for integers n,m ≥ 3. We apply similar techniques to present upper bounds for the power domination number of generalized Petersen graphs P(m,k). We prove those upper bounds provide the exact values of the power domination numbers if the integers m,n, and k satisfy some given relations. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 58(1), 43–49 2011