On the metric representation of the vertices of a graph

TL;DR

This paper characterizes which finite subsets of integer coordinate vectors can be realized as metric representations of vertices in a graph relative to a resolving set, and explores the uniqueness of such realizations especially in two dimensions.

Contribution

It provides a characterization of realizable metric representation sets and examines the role of strong path products in this context, including uniqueness in two dimensions.

Findings

Characterization of realizable sets in general dimensions.

Analysis of the role of strong product of paths.

Complete characterization of unique realizations in two dimensions.

Abstract

The metric representation of a vertex $u$ in a connected graph $G$ respect to an ordered vertex subset $W=\{\omega_1, \dots , \omega_n\}\subset V(G)$ is the vector of distances $r(u\vert W)=(d(u,\omega_1), \dots , d(u,\omega_n))$. A vertex subset $W$ is a resolving set of $G$ if $r(u\vert W)\neq r(v\vert W)$, for every $u,v\in V(G)$ with $u\neq v$. Thus, a resolving set with $n$ elements provides a set of metric representation vectors $S\subset \mathbb{Z}^n$ with cardinal equal to the order of the graph. In this paper, we address the reverse point of view, that is, we characterize the finite subsets $S\subset \mathbb{Z}^n$ that are realizable as the set of metric representation vectors of a graph $G$ with respect to some resolving set $W$. We also explore the role that the strong product of paths plays in this context. Moreover, in the case $n=2$, we characterize the sets $S\subset \mathbb{Z}^2$ that are uniquely realizable as the set of metric representation vectors of a graph $G$ with respect to a resolving set $W$.