Metric representations by minimal graphs
Víctor Franco-Sánchez, Mercè Mora, María Luz Puertas
TL;DR
The paper addresses which sets $S\subset \mathbb{Z}^n$ can be realized as the metric-coordinate image of a graph with a resolving set. It shows all realizations are spanning subgraphs of the canonical realization $\widehat{G}$ and derives exact edge-addition and edge-removal criteria, enabling construction of minimal and uniquely realizable representations. Computational results include a polynomial-time test for $METREL$ and an NP-completeness proof for $BMETREL$ via a polynomial reduction from $3$-SAT. For trees, the paper gives a complete characterization of tree realizations, constructs a canonical tree realization, and provides a uniqueness criterion showing that tree realizations are equivalent when applicable.
Abstract
A resolving set in a graph $G$ is a vertex subset $W= \{ω^1, \dots, ω^n\} \subseteq V(G)$ such that each $u \in V(G)$ can be uniquely identified by the vector $r(u \vert W) = (d(u,ω^1), \dots, d(u,ω^n))$ of metric coordinates of $u$ with respect to $W$. The reverse problem of identifying the vector sets that are a set of coordinates of some graph provides the concept of realizable vector set $S \subset \mathbb{Z}^n$ by a pair $(G, W)$ meaning that $S=\{ r(u\vert W)\colon u\in V(G)\}$ with $W$ a resolving set of the graph $G$. Here we focus on the minimality of the realizations of vector sets with respect to their edge sets. On the one hand, we study conditions under which it is possible to remove an edge from the graph and keep the realizability condition. This provides a method for finding minimal realizations, as well as allowing us to characterize uniquely realizable vector sets. On the other hand, we prove that the decision problem of realizing a vector set by a graph with a given number of edges is an NP-complete problem. Finally, we characterize the vector sets that are realizable by a tree and, furthermore, we study the case in which such a realization is the only one.