{
  "abstract": "In 2019, Sivaraman conjectured that every $P_k$-free graph has cop number at most $k-3$ for $k\\geq 5$. Chudnovsky, Norin, Seymour, and Turcotte proved $P_5$-free graphs have cop number at most $2$. For $k\\geq 6$ Sivaraman's conjecture remains widely open. There is little evidence to believe Sivaraman's conjecture is optimal, making the discovery of substantially better bounds interesting, even for subclasses of $P_k$-free graphs. We consider this problem for graphs with no long holes, which generalizes $P_k$-free graphs, that also have some forbidden shorter holes. Specifically, we show that for $t\\geq 13$, $(C_4,\\dots, C_{t}, C_k,C_{k+1},\\dots)$-free graphs have cop number at most $\\lceil \\frac{k-2}{\\lfloor \\frac{t}{4}\\rfloor }\\rceil+1$. Taking $C_{k-1}$, which has cop number $2$ whenever $k\\geq 5$, demonstrates that this bound is tight up to a small constant factor for infinity many pairs $k$ and $t$. The proof relies on showing that in graphs without short holes, the closed neighbourhood of a short induced path can be guarded by $1$ cop, even when these paths are not geodesics. Additionally, for $t\\geq 2$ we show that $(C_{t+2},C_{t+3},C_k, C_{k+1},\\dots, )$-free graphs have cop number at most $(t-1)\\lceil \\frac{(k-2)}{t}\\rceil$. This show that $(C_4,C_5,C_k,C_{k+1},\\dots)$-free graphs have cop number at most $\\lceil \\frac{k-2}{2}\\rceil$, extending work by Clow and Meger on $(P_k,\\text{claw}, \\text{butterfly}, C_4,C_5)$-free graphs to a substantially more general class.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36290",
  "author_total": 1,
  "authors": [
    {
      "name": "Alexander Clow"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3ke9y2bzcqh6m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36290",
  "fulltext_url": null,
  "id": "3ke9y2bzcqh6m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36290",
  "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": "W7224100128",
  "pdf_url": "https://arxiv.org/pdf/2610.36290",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3ke9y2bzcqh6m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36290",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36290"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Cops and Robbers on Graphs without Long or Short Holes",
  "title_text": "Cops and Robbers on Graphs without Long or Short Holes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/3ke9y2bzcqh6m",
  "venue": "arXiv",
  "versions": []
}