Metric Dimension Threshold of Graphs

Abstract

Let G be a connected graph. A subset S of vertices of G is said to be a resolving set of G, if for any two vertices u and v of G there is at least a member w of S such that du,w≠dv,w. The minimum number t that any subset S of vertices G with S=t is a resolving set for G, is called the metric dimension threshold, and is denoted by dimthG. In this paper, the concept of metric dimension threshold is introduced according to its application in some real-word problems. Also, the metric dimension threshold of some families of graphs and a characterization of graphs G of order n for which the metric dimension threshold equals 2, n−2, and n−1 are given. Moreover, some graphs with equal the metric dimension threshold and the standard metric dimension of graphs are presented.