Paths and Intersections: Minimum Realization of Okamura-Seymour Instances

Yu Chen; Pavlo Pylyavskyy; Zihan Tan

Paths and Intersections: Minimum Realization of Okamura-Seymour Instances

Data Structures and Algorithms 2026-07-03 v1 Computational Geometry Combinatorics

Authors: Yu Chen , Pavlo Pylyavskyy , Zihan Tan

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.

Keywords

Okamura-Seymour instances shortest-path metrics minimum realization medial graph graph algorithms inverse problem

Cite

@article{arxiv.2607.02883,
  title  = {Paths and Intersections: Minimum Realization of Okamura-Seymour Instances},
  author = {Yu Chen and Pavlo Pylyavskyy and Zihan Tan},
  journal= {arXiv preprint arXiv:2607.02883},
  year   = {2026}
}