Paths and Intersections: Recognizing Outerplanar Metrics

2026-06-24

AI Summary

This work addresses the problem of determining whether a given metric on a set of terminals can be exactly realized as the shortest-path distances in some edge-weighted outerplanar graph. The authors introduce a novel structural perspective based on “paths and their intersections,” departing from traditional frameworks that rely on local characterizations—such as those for tree-like metrics or the Okamura–Seymour setting—and prove that outerplanar metrics admit no local characterization. Building on this insight, they devise the first polynomial-time algorithm in the number of terminals to decide realizability, thereby enabling efficient recognition of outerplanar metrics.

Abstract

We study the following distance realization problem: given a metric $D$ on a set $T$ of terminals, does there exist an (edge-weighted) outerplanar graph $G$, such that $T\subseteq V(G)$, and for every pair $t,t'\in T$, $\textsf{dist}_G(t,t')=D(t,t')$? We first prove that there is no ``local characterization'', forming a contrast with trees and Okamura-Seymour instances. Our main result is an efficient algorithm for this problem whose running time is polynomial in $|T|$. Both our proof and our algorithm utilize a recent new approach of analyzing graph structures, by viewing graphs as paths and their intersections, which we believe is of independent interest.

Problem

• outerplanar graph
• distance realization
• metric embedding
• graph realization
• shortest path metric

Innovation

• outerplanar metrics
• distance realization
• path intersection representation
• polynomial-time algorithm
• graph characterization