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

Abstract

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, …, wk } 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, wk )). 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 γre (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 γreG of some comb product graphs in this paper, namely Pm ⊲ Pn , Sm ⊲ Pn , and Km ⊲ Pn .

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