Equitable Power Domination Number of Certain Graphs

Abstract

Let   12G">  be a graph with vertex set 12V"> , a set 12Sâٹ†V">  is said to be a power dominating set (PDS), if every vertex  12u∈V-S">  is observed by some vertices in 12S">  using the following rules: (i) if a vertex  12v">  in 12G">  is in PDS, then it dominates itself and all the adjacent vertices of 12v">  and (ii) if an observed vertex 12v">  in 12G">  has  12k>1">  adjacent vertices and if 12k-1">  of these vertices are already observed, then the remaining one non-observed vertex will also be observed by 12v">  in 12G"> . The degree 12d(v)">  of a vertex 12v">  in 12G">  is the number of edges of 12G">  incident with 12v">  and any two adjacent vertices 12u">  and 12v">  in 12G">  are said to hold equitable property if 12|d(u)-d(v)| ≤ 1"> . In this paper, we introduce the notions of equitable power dominating set and equitable power domination number. We also derive the equitable power domination number of certain graphs.