Title : ( Metric Dimension Threshold of Graphs )
Authors: Meysam Korivand , Kazem Khashyarmanesh , Mostafa Tavakoli ,Access to full-text not allowed by authors
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 $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\T{d}(u, w) \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\neq \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\T{d}(v, w)$. The minimum number $t$ that any subset $S$ of vertices $G$ with $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\vert S \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\vert =t$ is a resolving set for $G$, is called the metric dimension threshold and is denoted by $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\T{dim}_{th}(G)$. 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.