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 d u , w ≠ d v , 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 dim th G . 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.