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One-Vertex Lifts of Looped Circle Graphs: Obstructions and Interlace Polynomials — arXiv:2611.05961

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.05961
License (as archived): CC0-1.0
Archived: 2026-11-05T18:28:25.534Z
SHA-256 of the archived text: e2b5f98f9b7576f4124ff9e284e2f31d522b9d3e327a9dabaebd94be90b03b35
Stable URL of this page: https://metraipolis.com/documento/dc_596be442b5aa

How to cite

One-Vertex Lifts of Looped Circle Graphs: Obstructions and Interlace Polynomials — arXiv:2611.05961 — arXiv (en), https://arxiv.org/abs/2611.05961 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_596be442b5aa (Metraipolis archive dc_596be442b5aa, sha256 e2b5f98f9b7576f4124ff9e284e2f31d522b9d3e327a9dabaebd94be90b03b35)

BibTeX

@misc{dc596be442b5aa,
  title = {One-Vertex Lifts of Looped Circle Graphs: Obstructions and Interlace Polynomials — arXiv:2611.05961},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.05961},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_596be442b5aa; sha256 e2b5f98f9b7576f4124ff9e284e2f31d522b9d3e327a9dabaebd94be90b03b35},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.05961. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2611.05961 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-05T18:18:33.996Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).