Graphs with the edge metric dimension smaller than the metric dimension

TL;DR

This paper proves the existence of graphs where the edge metric dimension is strictly smaller than the metric dimension, resolving open problems and showing such graphs are infinitely many, with no constant factor bounds linking the two dimensions.

Contribution

It settles three open problems by demonstrating the unbounded difference between metric and edge metric dimensions in graphs.

Findings

Existence of graphs with arbitrarily large order where edge metric dimension is smaller than metric dimension.

No constant factor can bound the edge metric dimension by the metric dimension.

Infinitely many such graphs exist, disproving previous assumptions.

Abstract

Given a connected graph $G$, the metric (resp. edge metric) dimension of $G$ is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of $G$ by means of distance vectors to such a set. In this work, we settle three open problems on (edge) metric dimension of graphs. Specifically, we show that for every $r,t\ge 2$ with $r\ne t$, there is $n_0$, such that for every $n\ge n_0$ there exists a graph $G$ of order $n$ with metric dimension $r$ and edge metric dimension $t$, which among other consequences, shows the existence of infinitely many graph whose edge metric dimension is strictly smaller than its metric dimension. In addition, we also prove that it is not possible to bound the edge metric dimension of a graph $G$ by some constant factor of the metric dimension of $G$.