Metric Dimension Threshold of Graphs. (14th September 2022) - Korivand, Meysam; Khashyarmanesh, Kazem; Tavakoli, Mostafa
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.