On the power domination number of corona product and join graphs

Abstract

A set D of vertices of graph G is called a dominating set if every vertex is adjacent to some vertex . A set D to be a power dominating set of a graph if every vertex and every edge in the system are monitored by the set D. The power domination number γP (G) of a graph G is the minimum cardinality of a power dominating set of G. In this paper, we analyze the power domination number of corona graphs and join graphs. The corona product of two graphs G 1 and G 2 denoted by is defined as the graph G obtained by taking one copy of G 1 and copies of G 2, and then joined by an edge the i'th vertex of G 1 to every vertex in the i'th copy of G 2. The join of two graphs H 1 and H 2 is a graph formed from disjoint copies of H 1 and H 2 by connecting every vertex of H 1 to every vertex of H 2. Join graph H 1 and H 2 denoted by H 1 + H 2. The results show that the power domination number of some corona product and join graphs attain the lower bound.