{
  "abstract": "The Pachter-Sturmfels phylogenetic rank of a graph G is the minimal number of metric trees needed to embed G isometrically. Here, all edges of G are of length one and the product of metric trees is endowed with the supremum norm. We develop both a greedy and an exact algorithm for computing phylogenetic ranks. Using our algorithms, we construct a database of phylogenetic ranks which includes all graphs on 6 and 7 vertices. In particular, we exhibit examples disproving that the phylogenetic rank is hereditary, bounded by $\\lceil \\frac{n}{2} \\rceil$, and a generalised 4-point conjecture by Pachter and Sturmfels. In addition, we show that the phylogenetic rank is subadditive under 1-sums and certain 2-vertex-sums, and that it is trivially upper bounded by n-1, where n denotes the number of vertices. We also provide a complete classification of graphs with phylogenetic rank 1 and construct several infinite families with phylogenetic rank $\\lceil \\frac{n}{2} \\rceil$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19372",
  "author_total": 6,
  "authors": [
    {
      "name": "Franklin Ashworth"
    },
    {
      "name": "Oliver Clarke"
    },
    {
      "name": "Jeffrey Giansiracusa"
    },
    {
      "name": "Jackson Jones"
    },
    {
      "name": "Julio Quijas-Aceves"
    },
    {
      "name": "Yue Ren"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6dsxknkwm5sfc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19372",
  "fulltext_url": null,
  "id": "6dsxknkwm5sfc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19372",
  "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",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7213606536",
  "pdf_url": "https://arxiv.org/pdf/2609.19372",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6dsxknkwm5sfc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19372",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19372"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The phylogenetic rank of a graph",
  "title_text": "The phylogenetic rank of a graph",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/6dsxknkwm5sfc",
  "venue": "arXiv",
  "versions": []
}