Equitable power domination number of total graph of certain graphs

Abstract

Abstract Let G ( V , E ), or simply G , be a graph. A set S ⊆ V is said to be a power dominating set (PDS) if every vertex u ∈ V − S is observed by certain vertices in S by the following two rules: (a) if a vertex v in G is in PDS, then it dominates itself and all the adjacent vertices of v and (b) if an observed vertex v in G has k > 1 adjacent vertices and if k − 1 of these vertices are already observed, then the remaining one non-observed vertex is also observed by v in G. A power dominating set S ⊆ V in G is said to be an equitable power dominating set (EPDS), if for every vertex v ∈ V − S there exists an adjacent vertex u ∈ S such that | d ( u ) − d ( v )| ≤ 1, where d ( u ) and d ( v ) represents the degree of u and degree of v , respectively. The minimum cardinality of an EPDS of G is called the equitable power domination number (EPDN) of G , denoted by γ epd ( G ). The vertices and edges of G are called elements. Two elements of G are neighbors if they are either incident or adjacent in G . The total graph T ( G ) has vertex set V ( G ) ∪ E ( G ) and two vertices of T ( G ) are adjacent whenever they are neighbors in G . In this paper, we obtain the EPDN of the total graph of certain graphs.