Nonlocal Metric Dimension: Two Operations, Integer Linear Programming and an Application

Meysam Korivand

doi:10.1007/s40840-026-02092-8

Nonlocal Metric Dimension: Two Operations, Integer Linear Programming and an Application

2026-04-20

0

PRE

AI

DOI：10.1007/s40840-026-02092-8

Original

Original request for help

Save

Abstract

En 中文

The nonlocal metric dimension of a connected graph G, written as dimn & ell;(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ extrm{dim}_{n \ell }(G)$$\end{document}, is the smallest possible size of a set of vertices such that allows every two non-neighbor vertices to be distinguished by the distance of a vertex from that set. In this paper, we look at some unsolved issues concerning the nonlocal metric dimension of the corona product of two graphs. In particular, we demonstrate that dimn & ell;(T circle dot Km)=& ell;(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ extrm{dim}_{n \ell }(T \odot K_m) =\ell (T)$$\end{document}, where T is a tree with & ell;(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell (T)$$\end{document} leaves. In addition, for a connected bipartite graph G with partite sets A and B, we show that dim(G)+min{|A|,|B|}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ extrm{dim}(G) + \min \{|A|, |B|\}$$\end{document} is a sharp upper bound for dimn & ell;(G circle dot Km)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ extrm{dim}_{n \ell }(G \odot K_m)$$\end{document}. We also investigate the nonlocal metric dimension of the join graph G+K1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G+K_1$$\end{document} for certain graphs G, including the fan graph and a family of trees. Furthermore, a model based on integer linear programming is proposed for determining the nonlocal metric dimension. Finally, a real-world application of the nonlocal metric dimension is presented.

Keywords:

Nonlocal resolving set

Corona products

Join

Clique cover

Integer linear programming

Nonlocal Metric Dimension: Two Operations, Integer Linear Programming and an Application

Abstract

Journal

Organization

Cited Papers