On the edge dimension and fractional edge dimension of graphs

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.