Equitable power domination number of total graph of certain graphs

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 γepd (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.