{
  "abstract": "Let $M$ be the adjacency matrix of a looped simple graph over $\\mathbb{F}_2$ and let $t=\\operatorname{diag} M$. We study the one-vertex lift with adjacency matrix $\\left(\\begin{smallmatrix}M+tt^T&t\\\\t^T&0\\end{smallmatrix}\\right)$. A looped circle graph is called stable if its lift is a circle graph. We prove that stability is equivalent to the existence of a pseudo-orientable bouquet realizing the given looped circle graph. We then obtain a finite excluded toggle-minor characterization by applying two rooted constructions to the known excluded pivot-minors for circle graphs. An exact enumeration yields $22$ minimal toggle-equivalence classes: one of order $5$, twelve of order $6$, and nine of order $7$. The enumeration is accompanied by independently checkable linear-algebra and minor certificates. Finally, we prove $q(G^\\dagger;y)=yq(G;y)$ for the vertex-nullity interlace polynomial of every looped simple graph, and derive quasi-tree counting and real determinant formulas.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05961",
  "author_total": 2,
  "authors": [
    {
      "name": "Qingying Deng"
    },
    {
      "name": "Yexiang Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fj8z7mhwvaa0b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05961",
  "fulltext_url": null,
  "id": "fj8z7mhwvaa0b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05961",
  "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/2611.05961",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fj8z7mhwvaa0b.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.05961",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05961"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "One-Vertex Lifts of Looped Circle Graphs: Obstructions and Interlace Polynomials",
  "title_text": "One-Vertex Lifts of Looped Circle Graphs: Obstructions and Interlace Polynomials",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/fj8z7mhwvaa0b",
  "venue": "arXiv",
  "versions": []
}