{
  "abstract": "We study two inverse problems for shortest-path metrics of Okamura-Seymour instances: recognizing metrics realizable by outerplanar graphs, and computing minimum-edge Okamura-Seymour realizations. We introduce the notion of \\emph{repelling pairs}, a metric certificate that the shortest paths corresponding to two terminal pairs must be vertex-disjoint in every realization. Our central structural result is that, for an Okamura-Seymour metric with a prescribed cyclic order, a terminal path structure in an Okamura-Seymour instance can be realized by nonnegative edge lengths if and only if the paths assigned to every repelling pair are vertex-disjoint. Building on the notion of repelling pairs, we give algorithmic answers to the inverse problems. First, we design an algorithm that, given a metric, decides in polynomial time whether or it admits an outerplanar realization and constructs one when one exists. Second, given an Okamura-Seymour metric, we efficiently compute a canonical medial template whose crossing number equals the minimum number of edges in any Okamura-Seymour realization. The minimum-edge graph structures are exactly the primal graphs of arrangements of this template, and each can be assigned realizing edge lengths in polynomial time.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.02883",
  "author_total": 3,
  "authors": [
    {
      "name": "Yu Chen"
    },
    {
      "name": "Pavlo Pylyavskyy"
    },
    {
      "name": "Zihan Tan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/41deyf3p91vhk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.02883",
  "fulltext_url": null,
  "id": "41deyf3p91vhk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.02883",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CG"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7167577654",
  "pdf_url": "https://arxiv.org/pdf/2607.02883",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/41deyf3p91vhk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2607.02883",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.02883"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Paths and Intersections: Repelling Pairs",
  "title_text": "Paths and Intersections: Repelling Pairs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/41deyf3p91vhk",
  "venue": "arXiv",
  "versions": []
}