Nonlocal Metric Dimension of Windmill Graph

Abstract

Let G = (V (G), E(G)) be a simple and connected graph. The distance between two vertices u and v
in G, is the length of a shortest path from u to v, denoted by d(u, v). Suppose S = {s1, s2, ...sk} is an
ordered subset of vertices of G, then the metric representation of a vertex u ∈ V (G) with respect to S,
denoted by r(u|S), is the k−vector (d(u, s1), d(u, s2), ..., d(u, sk)). If every two nonadjacent vertices of
G have distinct metric representations with respect to S, then the set S is called a nonlocal resolving
set for G. A nonlocal resolving set with minimum cardinality is called a nonlocal metric basis. The
nonlocal metric dimension of G is the cardinality of the nonlocal metric basis of G and is denoted by
nldim(G). In this paper, we obtained nonlocal metric dimension of windmill graph.