{
  "abstract": "In 1959, Tutte proved that a matroid can be realised by a graph if and only if it does not have a minor isomorphic to $U_{2,4}$, $F_7$, $F_7^*$, $M^*(K_5)$ or $M^*(K_{3,3})$. In this paper, we extend this theorem to determine when it is possible to find a graphic realisation $G$ of a matroid $M$, subject to a list of additional constraints of the following form: a subset $F$ of the elements of $M$ must induce a connected subgraph of $G$. We prove that a connected matroid with such constraints---called a `constraint matroid'---is realisable as a graph satisfying those constraints if and only if it does not contain as a minor one of the constraint matroids from a given list: that is, the original $5$ matroids of Tutte, a further $9$ sporadic constraint matroids, and two well-behaved infinite families. Matroids realisable as graphs appear famously in Whitney's planarity criterion: Carmesin proved an extension of Whitney's planarity criterion to the embeddability of local simply connected $2$-complexes into $\\mathbb{R}^3$. An application of our theorem is an extension of this theorem beyond the simply connected regime.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01841",
  "author_total": 1,
  "authors": [
    {
      "name": "Will J. Turner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/61dym17kjjrsz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01841",
  "fulltext_url": null,
  "id": "61dym17kjjrsz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01841",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01841",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/61dym17kjjrsz.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01841",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01841"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Graphic matroids with connectivity constraints",
  "title_text": "Graphic matroids with connectivity constraints",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/61dym17kjjrsz",
  "venue": "arXiv",
  "versions": []
}