Uniquely identifying the edges of a graph: the edge metric dimension

TL;DR

This paper introduces the concept of edge metric dimension in graphs, explores its properties, compares it with the standard metric dimension, and discusses computational complexity and bounds for various graph classes.

Contribution

It formally defines the edge metric dimension, analyzes its properties, compares it with the standard metric dimension, and establishes complexity and bounds results.

Findings

Computing edge metric dimension is NP-hard.

Provides bounds and formulas for specific graph classes.

Shows relationships between edge and standard metric dimensions.

Abstract

Let $G=(V,E)$ be a connected graph, let $v\in V$ be a vertex and let $e=uw\in E$ be an edge. The distance between the vertex $v$ and the edge $e$ is given by $d_G(e,v)=\min\{d_G(u,v),d_G(w,v)\}$. A vertex $w\in V$ distinguishes two edges $e_1,e_2\in E$ if $d_G(w,e_1)\ne d_G(w,e_2)$. A set $S$ of vertices in a connected graph $G$ is an edge metric generator for $G$ if every two edges of $G$ are distinguished by some vertex of $S$. The smallest cardinality of an edge metric generator for $G$ is called the edge metric dimension and is denoted by $edim(G)$. In this article we introduce the concept of edge metric dimension and initiate the study of its mathematical properties. We make a comparison between the edge metric dimension and the standard metric dimension of graphs while presenting some realization results concerning the edge metric dimension and the standard metric dimension of graphs. We prove that computing the edge metric dimension of connected graphs is NP-hard and give some approximation results. Moreover, we present some bounds and closed formulae for the edge metric dimension of several classes of graphs.