Journal article
Electronic Journal of Graph Theory and Applications
• 2018

Let G=(V,E) be a simple connected graph. A subset S of V is called a neighbourhood set of G if $G=\bigcup_{s\in S}<N[s]>$, where N[v] denotes the closed neighbourhood of the vertex v in G. Further for each ordered subset S={s_1,s_2, ...,s_k} of V and a vertex $u\in V$, we associate a vector $\Gamma(u/S)=(d(u,s_1),d(u,s_2), ...,d(u,s_k))$ 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 $\Gamma(u/S)\neq \Gamma(v/S)$ for all $u,v\in V-S$. 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, particulary for paths and cycles.