On Edge Dimension of a Graph

TL;DR

This paper investigates the edge dimension of graphs, answering key questions about its properties, classifying certain graphs, and computing edge dimensions for specific graph operations.

Contribution

It classifies graphs with maximum edge dimension and explores the unbounded ratio between edge and metric dimensions, also computing edge dimensions for graph products and joins.

Findings

Classified graphs with edge dimension n-1

Showed ratio of edge to metric dimension is unbounded

Computed edge dimensions for graph products and joins

Abstract

Given a connected graph $G(V, E)$, the edge dimension, denoted $\mathrm{edim}(G)$, is the least size of a set $S \subseteq V$ that distinguishes every pair of edges of $G$, in the sense that the edges have pairwise distinct tuples of distances to the vertices of $S$. The notation was introduced by Kelenc, Tratnik, and Yero, and in their paper, they asked several questions about properties of $\mathrm{edim}$. In this article we answer two of these questions: we classify the graphs for which $\mathrm{edim}(G) = n-1$ and show that $\frac{\mathrm{edim}(G)}{\dim(G)}$ isn't bounded from above (here $\dim(G)$ is the standard metric dimension of $G$). We also compute $\mathrm{edim}(G\Box P_m)$ and $\mathrm{edim}(G + K_1)$.