On the edge dimension and fractional edge dimension of graphs

TL;DR

This paper introduces and explores the fractional edge dimension of graphs, providing theoretical results, constructions, and characterizations, including bounds, specific graph classes, and relationships with graph substructures.

Contribution

It defines fractional edge dimension, establishes bounds, constructs examples with large differences, and characterizes graphs with specific fractional edge dimensions.

Findings

Fractional edge dimension ranges from 1 to n/2 for connected graphs.

Graphs with fractional edge dimension 1 are characterized.

Existence of non-isomorphic graphs with identical edge metric coordinates.

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and let $d(u,w)$ denote the length of a $u-w$ geodesic in $G$. For any $v\in V(G)$ and $e=xy\in E(G)$, let $d(e,v)=\min\{d(x,v),d(y,v)\}$. For distinct $e_1, e_2\in E(G)$, let $R\{e_1,e_2\}=\{z\in V(G):d(z,e_1)\neq d(z,e_2)\}$. Kelenc et al. [Discrete Appl. Math. 251 (2018) 204-220] introduced the edge dimension of a graph: A vertex subset $S\subseteq V(G)$ is an edge resolving set of $G$ if $|S\cap R\{e_1,e_2\}|\ge 1$ for any distinct $e_1, e_2\in E(G)$, and the edge dimension $edim(G)$ of $G$ is the minimum cardinality among all edge resolving sets of $G$. For a real-valued function $g$ defined on $V(G)$ and for $U\subseteq V(G)$, let $g(U)=\sum_{s\in U}g(s)$. Then $g:V(G)\rightarrow[0,1]$ is an edge resolving function of $G$ if $g(R\{e_1,e_2\})\ge1$ for any distinct $e_1,e_2\in E(G)$. The fractional edge dimension $edim_f(G)$ of $G$ is $\min\{g(V(G)):g\mbox{ is an edge resolving function of }G\}$. Note that $edim_f(G)$ reduces to $edim(G)$ if the codomain of edge resolving functions is restricted to $\{0,1\}$. We introduce and study fractional edge dimension and obtain some general results on the edge dimension of graphs. We show that there exist two non-isomorphic graphs on the same vertex set with the same edge metric coordinates. We construct two graphs $G$ and $H$ such that $H \subset G$ and both $edim(H)-edim(G)$ and $edim_f(H)-edim_f(G)$ can be arbitrarily large. We show that a graph $G$ with $edim(G)=2$ cannot have $K_5$ or $K_{3,3}$ as a subgraph, and we construct a non-planar graph $H$ satisfying $edim(H)=2$. It is easy to see that, for any connected graph $G$ of order $n\ge3$, $1\le edim_f(G) \le \frac{n}{2}$; we characterize graphs $G$ satisfying $edim_f(G)=1$ and examine some graph classes satisfying $edim_f(G)=\frac{n}{2}$. We also determine the fractional edge dimension for some classes of graphs.