On classes of neighborhood resolving sets of a graph

Abstract

Let G = (V, E) be a simple connected graph. A subset S of V is called a neighbourhood set of G if G = ⋃_s ∈ S < N[s] > , where N[v] denotes the closed neighbourhood of the vertex v in G. Further for each ordered subset S = {s₁, s₂, ..., s_k} of V and a vertex u ∈ V, we associate a vector Γ(u/S) = (d(u, s₁), d(u, s₂), ..., d(u, s_k)) with respect to S, where d(u, v) denote the distance between u and v in G. A subset S is said to be resolving set of G if Γ(u/S) ≠ Γ(v/S) for all u, v ∈ V − S. A neighbouring set of G which is also a resolving set for G is called a neighbourhood resolving set (nr-set). The purpose of this paper is to introduce various types of nr-sets and compute minimum cardinality of each set, in possible cases, particularly for paths and cycles.