On resolving efficient domination number of path and comb product of special graph

We use finite, connected, and undirected graph denoted by G . Let V ( G ) and E ( G ) be a vertex set and edge set respectively. A subset D of V ( G ) is an efficient dominating set of graph G if each vertex in G is either in D or adjoining to a vertex in D . A subset W of V ( G ) is a resolving set of G if any vertex in G is differently distinguished by its representation respect of every vertex in an ordered set W . Let W = { w 1 , w 2 , w 3 , …, w k } be a subset of V ( G ). The representation of vertex υ ∈ G in respect of an ordered set W is r ( υ | W ) = ( d ( υ, w 1 ), d ( υ, w 2 ), …, d ( υ, w k )). The set W is called a resolving set of G if r ( u | W ) ≠ r ( υ | W ) ∀ u, υ ∈ G . A subset Z of V ( G ) is called the resolving efficient dominating set of graph G if it is an efficient dominating set and r ( u | Z ) ≠ r ( υ | Z ) ∀ u, υ ∈ G . Suppose γ re ( G ) denotes the minimum cardinality of the resolving efficient dominating set. In other word we call a resolving efficient domination number of graphs. We obtained γ re G of some comb product graphs in this paper, namely P m ⊲ P n , S m ⊲ P n , and K m ⊲ P n .