{
  "abstract": "For a planar graph \\(G\\), a plane subgraph \\(H\\) is plane-saturated with respect to \\(G\\) if adding any edge to the fixed drawing of \\(H\\), possibly together with new vertices, either creates a crossing or results in a graph that is not a subgraph of \\(G\\). The plane-saturation ratio \\(\\psr(G)\\) is the minimum of \\(e(H)/e(G)\\) over all plane-saturated subgraphs \\(H\\) of \\(G\\). For nonnegative integers \\(k_1\\) and \\(k_2\\), let \\(\\cG_{k_1,k_2}\\) denote the class of planar graphs in which, for each \\(i\\in\\{1,2\\}\\), at most \\(k_i\\) vertices of degree \\(i\\) have any prescribed neighborhood, where \\(k_i=0\\) means that vertices of degree \\(i\\) are forbidden. We prove that every \\(G\\in\\cG_{k_1,k_2}\\) with at least one edge satisfies $\\psr(G)>\\frac{1}{9+k_1+6k_2}.$ This settles two conjectures of Clifton and Salia.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34407",
  "author_total": 2,
  "authors": [
    {
      "name": "Qing Cui"
    },
    {
      "name": "Shengjin Ji"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fsz00mjp32h72.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34407",
  "fulltext_url": null,
  "id": "fsz00mjp32h72",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34407",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7223853217",
  "pdf_url": "https://arxiv.org/pdf/2610.34407",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fsz00mjp32h72.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34407",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34407"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Plane-Saturation Ratio of Planar Graphs",
  "title_text": "The Plane-Saturation Ratio of Planar Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/fsz00mjp32h72",
  "venue": "arXiv",
  "versions": []
}