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 G1 and G2 denoted by is defined as the graph G obtained by taking one copy of G1 and copies of G2, and then joined by an edge the i’th vertex of G1 to every vertex in the i’th copy of G2. The join of two graphs H1 and H2 is a graph formed from disjoint copies of H1 and H2 by connecting every vertex of H1 to every vertex of H2. Join graph H1 and H2 denoted by H1 + H2. The results show that the power domination number of some corona product and join graphs attain the lower bound.