On classes of neighborhood resolving sets of a graph

B. Sooryanarayana (Department of Mathematical and Computational Studies, Dr. Ambedkar Institute of Technology, Bengaluru, Karnataka State, India)
Suma A. S. (Department of Mathematical and Computational Studies, Dr. Ambedkar Institute of Technology, Bengaluru, Karnataka State, India)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 6, No 1 (2018): Electronic Journal of Graph Theory and Applications

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, ..., sk} of V and a vertex u ∈ V, we associate a vector Γ(u/S) = (d(u, s1), d(u, s2), ..., d(u, sk)) 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.