Journal of Mathematics, Volume (2022), No (1), Year (2022-9) , Pages (1-6)

#### Title : ( Metric Dimension Threshold of Graphs )

Authors: Meysam Korivand , Kazem Khashyarmanesh , Mostafa Tavakoli ,

Citation: BibTeX | EndNote

#### 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.

#### Keywords

, Graph, metric dimension threshold.
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@article{paperid:1091428,
author = {Korivand, Meysam and Khashyarmanesh, Kazem and Tavakoli, Mostafa},
title = {Metric Dimension Threshold of Graphs},
journal = {Journal of Mathematics},
year = {2022},
volume = {2022},
number = {1},
month = {September},
issn = {2314-4629},
pages = {1--6},
numpages = {5},
keywords = {Graph; metric dimension threshold.},
}