{
  "abstract": "For a nonempty acyclic oriented graph $H$, let $p(H)$ be the order of a longest directed path and let $τ(H)$ be the least positive integer $n$ such that $H$ admits a homomorphism to every tournament of order $n$. For all $p\\ge3$ and $g\\ge1$, we construct a connected acyclic oriented graph $H$ with underlying girth greater than $g$, absolute and relative oriented clique numbers equal to three, and \\[ p(H)=p,\\qquad τ(H)=r_{\\mathrm{tr}}(p), \\] where $r_{\\mathrm{tr}}(p)=2^{Θ(p)}$ is the tournament Ramsey number for a transitive $p$-vertex tournament. This disproves the conjectured polynomial bounds under bounded absolute or relative oriented clique number. It also shows that a forbidden graph can yield a polynomially $τ$-bounded class only if its underlying graph is a forest. For fixed $g$, the least order of these examples is bounded by a polynomial in $p$. A separate construction gives maximum in- and outdegree $O(p^2)$, uniformly in $g$. For $p=4$, the least order is $2^{Θ(g)}$. We also establish polynomial $τ$-boundedness for every orientation of the two four-vertex trees. The pure-claw case follows from the known $O(p^4)$ bound. We obtain the bound $2p-2$ for mixed claws and one-turn orientations of $P_4$ when $p\\ge2$, and bounds $4$ and $3p-2$ for the directed and alternating orientations of $P_4$, respectively. In the alternating case, $τ(H)=p(H)$ when the underlying graph is triangle-free.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33600",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhenhua Lyu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4h0xmdtw2ac4q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33600",
  "fulltext_url": null,
  "id": "4h0xmdtw2ac4q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33600",
  "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": "W7214721595",
  "pdf_url": "https://arxiv.org/pdf/2609.33600",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4h0xmdtw2ac4q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33600",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33600"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Polynomial Compressibility and Forbidden Oriented Forests",
  "title_text": "Polynomial Compressibility and Forbidden Oriented Forests",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/4h0xmdtw2ac4q",
  "venue": "arXiv",
  "versions": []
}