{
  "abstract": "In 2019, Liu proved that the cop number of any $(P_k, claw)$-free graph is at most $k-3$. In 2026, Clow and Meger improved and generalized the result of Liu by showing that every connected $(P_k, E)$-free graph has cop number at most $\\lceil \\frac{k-1}{2} \\rceil + 3$, where the graph $E$ is obtained by subdividing two edges of a claw. In this paper, we show that every $(P_k, claw)$-free graph $G$ has cop number at most $\\lceil \\frac{k-1}{2} \\rceil$. This improves the result of Clow and Meger on claw-free graphs. We further show that this bound can be improved to $\\lceil \\frac{k}{3}\\rceil$, when $G$ contains a hole of length at least $k-1$. We show that the cop number of any $(P_k, fork)$-free graph is at most $\\lceil \\frac{2(k-1)}{3} \\rceil$. This also improves the result of Clow and Meger on fork-free graphs for some values of $k$. We further show that every $(P_7, fork)$-free graph has cop number at most $3$, and this bound is tight.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20129",
  "author_total": 3,
  "authors": [
    {
      "name": "Uttam K. Gupta"
    },
    {
      "name": "Dinabandhu Pradhan"
    },
    {
      "name": "Rashmi Rekha Swain"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7e8w9h7p016zj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20129",
  "fulltext_url": null,
  "id": "7e8w9h7p016zj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20129",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7222171553",
  "pdf_url": "https://arxiv.org/pdf/2610.20129",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7e8w9h7p016zj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20129",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20129"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Cop number of $P_k$-free graphs without a claw or a fork",
  "title_text": "Cop number of P_k-free graphs without a claw or a fork",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/7e8w9h7p016zj",
  "venue": "arXiv",
  "versions": []
}