Equitable Power Domination Number of Mycielskian of Certain Graphs

Abstract

Let  be a simple graph with vertex set  and edge set . A set  is called a power dominating set (PDS), if every vertex   is observed by some vertices in  by using the following rules: (i) if a vertex  in  is in PDS, then it dominates itself and all the adjacent vertices of  and (ii) if an observed vertex  in   has  adjacent vertices and if   of these vertices are already observed, then the remaining one non-observed vertex is also observed by  in . A power dominating set    in   is said to be an equitable power dominating set (EPDS), if for every  there exists an adjacent vertex   such that the difference between the degree of  and degree of  is less than or equal to 1, i.e., . The minimum cardinality of an equitable power dominating set of  is called the equitable power domination number of  and denoted by . The Mycielskian of a graph  is the graph  with vertex set  where , and edge set  In this paper we investigate the equitable power domination number of Mycielskian of certain graphs.