On the uniqueness of continuation of a partially defined metric

TL;DR

This paper investigates the conditions under which a partially defined metric on a graph can be uniquely extended to a full metric, using graph theory to establish necessary and sufficient criteria.

Contribution

It provides a necessary and sufficient condition for the unique continuation of a partially defined metric on a graph.

Findings

Identifies conditions for unique metric extension

Uses graph theory to analyze metric continuation

Establishes a criterion for metric uniqueness

Abstract

The problem of continuation of a partially defined metric can be efficiently studied using graph theory. Let $G=G(V,E)$ be an undirected graph with the set of vertices $V$ and the set of edges $E$. A necessary and sufficient condition under which the weight $w\colon E\to\mathbb R^+$ on the graph $G$ has a unique continuation to a metric $d\colon V\times V\to\mathbb R^+$ is found.