ON THE NUMBER OF DOMINATING SETS OF POWER GRAPHS

Let G = (V, E) be a simple graph. A set S ⊆ V is a dominating set if every vertex in V \S is adjacent to at least one vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set in G. The m th power of G, is a graph with same set of vertices of G and an edge between two vertices if and only if there is a path of length at most m between them. For any n ∈ N , the n-subdivision of G is a simple graph G 1 n which is constructed by replacing each edge of G with a path of length n. The fractional power of G, denoted by G m n is m-th power of the n-subdivision of G or n-subdivision of m-th power of G. In this paper we study the number of dominating sets of power of certain graphs.