{
  "abstract": "Cizma and Linial asked for a classification of the metrizable theta graphs. We solve both their problem and its strict analogue. For $a\\le b\\le c$, the theta graph $Θ_{a,b,c}$ is metrizable if and only if $a\\le 2$ or $(a,b,c)=(3,3,3)$, and it is strictly metrizable if and only if $a\\le2$. Thus $Θ_{3,3,3}$ is precisely the exceptional theta graph that is metrizable but not strictly metrizable. The negative directions follow from the known obstructions $Θ_{3,3,4}$ and $Θ_{3,3,3}$ together with topological-minor closure. The positive direction is constructive. Consistency turns the possible detours through a length-two arm into compatible Ferrers relations, which are represented by one-dimensional potentials; all resulting shortest-path comparisons have positive slack. The exceptional ordinary-metrizable graph $Θ_{3,3,3}$ is handled by a two-threshold weak Ferrers representation. The proof is structural, yields rational edge lengths algorithmically, and uses no enumeration of path systems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.16357",
  "author_total": 1,
  "authors": [
    {
      "name": "Guangfu Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fg86c20dc1v5n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.16357",
  "fulltext_url": null,
  "id": "fg86c20dc1v5n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.16357",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7164935949",
  "pdf_url": "https://arxiv.org/pdf/2606.16357",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fg86c20dc1v5n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2606.16357",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.16357"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A complete classification of metrizable and strictly metrizable theta graphs",
  "title_text": "A complete classification of metrizable and strictly metrizable theta graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/fg86c20dc1v5n",
  "venue": "arXiv",
  "versions": []
}