Neighborhood Resolving Sets of a Graph

摘要

Let G = (V,E) be a simple connected graph. A subset S of V is called a neighborhood set of G if G =U_(s∈S), where N[v] denotes the closed neighborhood of the vertex v in G. Further, for each ordered subset S = (s_1, s_2,..., s_k) of V and a vertex u ∈ V, we associate a vector Γ(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 a resolving set of G if Γ(u/S) ≠ Γ(v/S) for all u, v ∈ V - S. A neighboring set of G which is also a resolving set for G is called a neighborhood resolving set (nr-set). The minimum cardinality of an nr-set of G is called the neighborhood metric dimension of G and is denoted by nmd(G). An nr-set with minimum cardinality is called an nrbasis. In this paper, we find the neighborhood metric dimension of some derived graphs of a path.