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 $\text{d}\left(u,w\right)\ne \text{d}\left(v,w\right)$ . The minimum number $t$ that any subset $S$ of vertices $G$ with $\left|S\right|=t$ is a resolving set for $G$ , is called the metric dimension threshold, and is denoted by ${\text{dim}}_{\text{th}}\left(G\right)$ . 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.