On resolving efficient domination number of comb product of special graphs

Abstract

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 ,…, w k } is the k –vector r ( υ | W ) = ( d ( υ , w 1 ), d ( υ , w 2 ),…, d ( v, w k )), 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 γ re ( 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 K n ▹ C 3 , K n ▹ P 3 , W n ▹ C 3 , W n ▹ P 3 , and S n ▹ P 2