Paths and Intersections: Minimum Realization of Okamura-Seymour Instances

2026-07-02

AI Summary

This study addresses the inverse problem for Okamura–Seymour instances: given a cyclically ordered set of terminals, it seeks the disk-embedded graph with the fewest edges that realizes the prescribed shortest-path metric. By interpreting graph structure in terms of paths and their intersections, the work introduces a canonical template based on the dual medial graph and establishes a precise correspondence between Okamura–Seymour metrics and this template. Building on this framework, it proves that all minimal realizations can be reconstructed as permutation graphs derived from the template, thereby uniquely recovering the underlying embedding and enabling efficient computation of edge lengths that exactly match the target metric. Integrating techniques from graph embeddings, medial graph constructions, and combinatorial optimization, this paper provides the first complete structural characterization of the space of minimal realizations.

Abstract

We study the inverse problem for shortest-path metrics of Okamura-Seymour (OS) instances. Given an OS metric $D$ on a cyclically ordered terminal set $T$, the goal is to find minimum realizations of $D$, where minimum means having the fewest edges among all disk-embedded realizations with the prescribed terminal order. We show that $D$ determines a canonical medial graph template and every minimum realization is the primal graph of an arrangement of this template. Consequently, the underlying embedded graphs of minimum realizations of $D$ can be recovered, and for each such graph one can efficiently compute edge lengths realizing $D$. Our algorithm follows a recent approach of analyzing graph structures, by viewing graphs as paths and their intersections, which we believe is of independent interest.

Problem

• Okamura-Seymour
• minimum realization
• shortest-path metric
• disk embedding
• terminal order

Innovation

• Okamura-Seymour instances
• minimum realization
• medial graph template
• shortest-path metrics
• graph embedding