{
  "abstract": "We study $W$-free planar graphs for $W\\in\\{W_{2,4},W_{2,5},W_{3,4}\\}$, where the quasi-double star $W_{h,k}$ is obtained from a three-vertex path by attaching $h$ leaves to one endpoint and $k$ leaves to the other. We prove that every $W_{2,4}$-free planar graph on $n$ vertices has at most $9n/4$ edges, and the bound is attained whenever $8\\mid n$. This determines the planar Turán density of $W_{2,4}$ as $9/4$. We also establish the sharp upper bound $5n/2$ for $W_{2,5}$. Combined with known constructions of planar graphs of maximum degree five, it yields $\\ex_{\\PP}(n,W_{2,5})=\\lfloor5n/2\\rfloor$ for every $n\\ge15$. These results close the two corresponding coefficient gaps in the bounds of Liu et~al. Our proofs use structural restrictions on high-degree vertices, local deletions, and degree deficits in neighborhoods of radius two. For $W_{3,4}$, we characterize the planar graphs with a dominating vertex that avoid this tree and determine their exact extremal number, $\\lfloor(5n-7)/2\\rfloor$, for every $n\\ge10$. Finally, $W$-free planar triangulations have at most eight, twelve, and eleven vertices for $W=W_{2,4},W_{2,5},W_{3,4}$, respectively; the first two bounds are sharp.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05706",
  "author_total": 3,
  "authors": [
    {
      "name": "Zehui Shao"
    },
    {
      "name": "Enqiang Zhu"
    },
    {
      "name": "Shaohui Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/40b45p2dmafyr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05706",
  "fulltext_url": null,
  "id": "40b45p2dmafyr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05706",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7220724462",
  "pdf_url": "https://arxiv.org/pdf/2610.05706",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/40b45p2dmafyr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05706",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05706"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sharp planar Turán bounds for quasi-double stars",
  "title_text": "Sharp planar Turán bounds for quasi-double stars",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/40b45p2dmafyr",
  "venue": "arXiv",
  "versions": []
}