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On the uniqueness of continuation of a partially defined metric

Evgeniy Petrov

TL;DR

The paper studies when a partially defined edge weight on a graph can be uniquely extended to a metric on the vertex set. It formalizes this extension problem via the pseudometrizable weight concept and the set $\mathfrak M_w$ of admissible extensions, with the weighted shortest-path pseudometric $d_w$ as the greatest element. The main contribution is a precise path-sequence criterion: the continuation to a metric is unique precisely when, for every non-adjacent pair with $d_w(u,v)>0$, there exists a sequence of $u$–$v$ paths $(P_n)$ satisfying $2\max_{e\in P_n} w(e) - w(P_n) \to d_w(u,v)$, which also forces the unique continuation to coincide with $d_w$. If the criterion fails for some pair, the authors construct another continuation, establishing non-uniqueness, and provide finite and infinite graph examples to illustrate both cases. This yields a complete characterization of uniqueness versus non-uniqueness in metric continuations for weighted graphs and connects to broader themes in metric extension theory on graphs.

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.

On the uniqueness of continuation of a partially defined metric

TL;DR

The paper studies when a partially defined edge weight on a graph can be uniquely extended to a metric on the vertex set. It formalizes this extension problem via the pseudometrizable weight concept and the set of admissible extensions, with the weighted shortest-path pseudometric as the greatest element. The main contribution is a precise path-sequence criterion: the continuation to a metric is unique precisely when, for every non-adjacent pair with , there exists a sequence of – paths satisfying , which also forces the unique continuation to coincide with . If the criterion fails for some pair, the authors construct another continuation, establishing non-uniqueness, and provide finite and infinite graph examples to illustrate both cases. This yields a complete characterization of uniqueness versus non-uniqueness in metric continuations for weighted graphs and connects to broader themes in metric extension theory on graphs.

Abstract

The problem of continuation of a partially defined metric can be efficiently studied using graph theory. Let be an undirected graph with the set of vertices and the set of edges . A necessary and sufficient condition under which the weight on the graph has a unique continuation to a metric is found.
Paper Structure (2 sections, 4 theorems, 27 equations, 2 figures)

This paper contains 2 sections, 4 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Uniqueness of continuation

Key Result

Theorem 1.4

Let $(G,w)$ be a weighted graph. The following statements are equivalent.

Figures (2)

  • Figure 1: An example of a finite weighted graph $(G,w)$ with the unique continuation.
  • Figure 2: A weighted graph $(G,w)$ with $\mathop{\mathrm{card}}\nolimits(V(G))=\aleph_0$.

Theorems & Definitions (8)

  • Theorem 1.4: DMV
  • Definition 1.7
  • Proposition 1.9: DMV
  • Theorem 2.1
  • proof
  • Corollary 2.11
  • Example 2.12
  • Example 2.13