{
  "abstract": "The 2-switch-degree $\\text{deg}(G)$ of a graph $G$ is the number of 2-switches that can be performed on $G$; equivalently, it is the degree of $G$ as a vertex of the realization graph $\\mathcal{G}(d)$ of its degree sequence $d$. We classify the prime graphs of 2-switch-degree at most 4, where a graph is prime if it is indecomposable with respect to the Tyshkevich composition and every vertex takes part in some 2-switch. From this classification we derive a sharp dichotomy that recovers the global shape of $\\mathcal{G}(d)$ from a single one of its local degrees: if $d$ has a realization $X$ with $\\text{deg}(X)=k\\le3$, then $\\mathcal{G}(d)$ is vertex-transitive and $k$-regular if and only if neither $T_{221}$ nor $\\overline{T_{221}}$ is an induced subgraph of $X$. Moreover, for every $k\\geq 4$ some prime graph of degree $k$ carries a 2-switch raising its degree to $2k-2$. As a further consequence, up to isomorphism there are only 10 realization graphs of prime graphs with degree at most 4.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40274",
  "author_total": 1,
  "authors": [
    {
      "name": "Victor N. Schvöllner"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1zrwdr1yheg62.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40274",
  "fulltext_url": null,
  "id": "1zrwdr1yheg62",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40274",
  "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"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7215125018",
  "pdf_url": "https://arxiv.org/pdf/2609.40274",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1zrwdr1yheg62.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.40274",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40274"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Classification of prime graphs with 2-switch-degree at most 4",
  "title_text": "Classification of prime graphs with 2-switch-degree at most 4",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/1zrwdr1yheg62",
  "venue": "arXiv",
  "versions": []
}