Equitable power domination number of the degree splitting graph of certain graphs

Abstract

For a graph G(V,E) with vertex set V and edge set E, a set S ⊆ V is called a power dominating set (PDS), if every vertex u ∈ V - S is observed by some vertices in S using the following rules: (i) if a vertex v in G is in PDS, then it dominates itself and all the adjacent vertices of v and (ii) 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 of G (V, E) is said to be an equitable power dominating set, if for every vertex v ∈ V - S there exists an adjacent vertex u ∈ S such that the difference between degree of u and degree of v is less than or equal to 1, that is |d(u) - d(v)| ≤ 1. The minimum cardinality of an equitable power dominating set of G is called the equitable power domination number of G and is denoted by γepd (G). Let G = (V, E) be a graph with V = S1 ∪ S2 ∪ …St ∪ T where each Si is a set of vertices having cardinality of at least two and the vertices are of the same degree and T = V\ ∪ St. The degree splitting graph of G is obtained from G by adding vertices w1, w2, …, wt and joining wi to each vertex of Si (1 ≤ i ≤ t) and is denoted by DS(G). In this paper we establish the equitable power domination number of the degree splitting graph of certain graphs.