On resolving efficient domination number of comb product of special graphs

Abstract

Let G be a connected, finite, and undirected graph. A vertex set D in G is an efficient dominating set of G if D is an independent set and for each point υ ∈ V(G)-D is adjacent to precisely one vertex d ∈ D. The representation of points υ ∈ V(G) in respect of an ordered set W = {w 1, w 2,…, wk } is the k–vector r(υ| W) = (d(υ, w 1), d(υ, w 2),…, d(v, wk )), which d(u, v) is the distance between the points u and υ. The set W is a resolving set of G if r(u|W) = r(υ| W), for each point u and υ in G. A set of vertices in graph G which is an efficient dominating set and resolving set is called a resolving efficient dominating set. The minimal cardinality of resolving efficient dominating set is called resolving efficient domination number, denoted by γre (G). The comb product between graph G and graph H is a graph which get from taking a copy of graph G as many vertices of graph H and grafting the i-th copy of graph G to each vertex of H, and its notated by G ▹ H. In this paper, we determine the resolving efficient domination number of comb product graph, namely Kn ▹C 3, Kn ▹P 3, Wn ▹C 3, Wn ▹P 3, and Sn ▹P 2