{
  "abstract": "\\textsc{Odd Cycle Transversal} is a classic $\\mathsf{NP}$-hard graph optimization problem asking for a minimum-weight set of vertices whose deletion makes the input graph bipartite, or equivalently, a maximum-weight induced bipartite subgraph. We show that \\textsc{Odd Cycle Transversal} is quasi-polynomial-time solvable on $kP_4$-free graphs, for every fixed $k \\in \\mathbb{N}$. In fact, we provide an $n^{O_k(\\log n)}$-time algorithm for the more general \\textsc{Max-Weight List $2$-Colorable Induced Subgraph}, where the notation $O_{k}(\\cdot)$ hides factors depending on $k$. Paired with known results from the literature, this allows us to obtain a complete complexity dichotomy for these two problems on $H$-free graphs into cases solvable in quasi-polynomial time and cases which are $\\mathsf{NP}$-hard, in particular resolving an open problem of Agrawal, Lima, Lokshtanov, Saurabh, and Sharma [SODA 2024]. Our algorithms are based on a new structural tool that may be of independent interest. We introduce the notion of $H$-amiable family and show that, for every fixed graph $H$ without isolated vertices and every fixed $k\\ge2$, every $kH$-free graph admits an $H$-amiable family of quasi-polynomial size that can be constructed in quasi-polynomial time. Besides yielding the aforementioned algorithms, this result gives, for every fixed connected graph $H$ and every fixed $k\\ge2$, a reduction from \\textsc{Max-Weight Independent Set} on $kH$-free graphs to the same problem on $H$-free graphs with $n^{O_{H,k}(\\log n)}$ overhead. In this setting, it improves the $n^{O_{H,k}(\\log^3 n)}$ overhead obtained by specializing the general reduction of Gartland and Lokshtanov [FOCS 2020].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30900",
  "author_total": 4,
  "authors": [
    {
      "name": "Esther Galby"
    },
    {
      "name": "Paloma T. de Lima"
    },
    {
      "name": "Andrea Munaro"
    },
    {
      "name": "Amir Nikabadi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fp5kaexkyhst2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30900",
  "fulltext_url": null,
  "id": "fp5kaexkyhst2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30900",
  "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": "W7214672918",
  "pdf_url": "https://arxiv.org/pdf/2609.30900",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fp5kaexkyhst2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.30900",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30900"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Odd Cycle Transversal on $H$-free graphs",
  "title_text": "Odd Cycle Transversal on H-free graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/fp5kaexkyhst2",
  "venue": "arXiv",
  "versions": []
}