{
  "abstract": "Let $X$ be a nodal curve, and $G$ be the graph dual to $X$. A stability condition on a graph $G$ is an assignment of integers to the biconnected subsets of vertices of $G$ satisfying some desired properties. Stability conditions yield degeneracy sets, which are certain collections of biconnected subsets of $G$, but degeneracy sets can also be described without a reference stability condition. We show that the degeneracy sets of a graph $G$ correspond to the single-element extensions of the graphic matroid $M(G)$. Some single-element extensions of the graphic matroid $M(G)$ can be oriented to be single-element extensions of the oriented graphic matroid $\\mathcal{M}(G)$. Single-element extensions of oriented matroids are in bijection with subdivisions of the Lawrence polytope of the dual oriented matroid. We construct stability conditions on $G$ from subdivisions of the Lawrence polytope of the cographic matroid $\\mathcal{M}^\\ast(G)$. Finally, we show that any degeneracy set $\\mathcal{D}$ of a graph $G$ that has the set of all biconnected subsets of $G$ as a lower bound in the poset of degeneracy sets corresponds to an orientable extension of $M(G)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00591",
  "author_total": 1,
  "authors": [
    {
      "name": "Natasha Crepeau"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6w69bvne6hwy1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00591",
  "fulltext_url": null,
  "id": "6w69bvne6hwy1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00591",
  "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/2610.00591",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6w69bvne6hwy1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00591",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00591"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Stability conditions and subdivisions of Lawrence polytopes",
  "title_text": "Stability conditions and subdivisions of Lawrence polytopes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/6w69bvne6hwy1",
  "venue": "arXiv",
  "versions": []
}