{
  "abstract": "Brambles of constant congestion in directed graphs have interesting structural and algorithmic applications, and it is desirable to find the smallest function $f_c(k)$ such that if a digraph has directed treewidth at least $f_c(k)$, then it contains a bramble of size at least $k$ and congestion at most $c$. By the Directed Grid Theorem of Kawarabayashi and Kreutzer [STOC 2015], such a function $f_c(k)$ exists for every value $c \\geq 2$ of the congestion. However, even using the recent improvements due to Hatzel et al. [FOCS 2024], the resulting function is a power tower of height $22$. Recently, Masařík et al. [SIAM J. Discrete Math. 2022] proved that, for congestion $8$, the function $f_8(k)$ can be made polynomial; more precisely, they showed that $f_8(k) = \\mathcal{O}(k^{48} \\cdot \\log^{13} k)$. They asked whether it is possible to decrease the congestion, ideally to $2$, even at the cost of a higher polynomial function. In this work we answer their question affirmatively with the best possible value of the congestion, by proving that $f_2(k) = \\mathcal{O}(k^{96}\\cdot \\log^{23} k)$. We also obtain smaller polynomials for higher congestion, namely $f_4(k) = \\mathcal{O}(k^{72} \\cdot \\log^{21} k)$ and $f_6(k) = \\mathcal{O}(k^{66} \\cdot \\log^{18} k)$. Our results are inspired by the proof for congestion $8$, which we improve by introducing walkages. Walkages allow us to avoid technicalities of the original proof by translating ideas due to Reed and Wood [Eur. J. Comb. 2012] that work for undirected graphs to the directed setting. Our proofs are constructive and naturally yield polynomial-time algorithms to find the brambles assuming a certificate for large directed treewidth is given. Moreover, we discuss algorithmic applications of our results for the Directed Disjoint Paths with Congestion problem, and to half-integral packings of subgraphs with large directed treewidth.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13270",
  "author_total": 4,
  "authors": [
    {
      "name": "Victor A. Campos"
    },
    {
      "name": "Raul Lopes"
    },
    {
      "name": "Rodrigo Nogueira"
    },
    {
      "name": "Ignasi Sau"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7s6gjb4n50rw7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13270",
  "fulltext_url": null,
  "id": "7s6gjb4n50rw7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13270",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7221548999",
  "pdf_url": "https://arxiv.org/pdf/2610.13270",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7s6gjb4n50rw7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13270",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13270"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Polynomial directed treewidth forces a large bramble with congestion two",
  "title_text": "Polynomial directed treewidth forces a large bramble with congestion two",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/7s6gjb4n50rw7",
  "venue": "arXiv",
  "versions": []
}