{
  "abstract": "A quasi-kernel of a digraph $D$ is an independent set from which every vertex is reachable by a directed path of length at most two. The Small Quasi-kernel Conjecture asserts that every source-free digraph on $n$ vertices has a quasi-kernel of size at most $n/2$, whereas the Large Quasi-kernel Conjecture asks for a quasi-kernel whose closed out-neighbourhood has size at least $n/2$. Spiro proved that the former implies the latter. In this paper, we establish a weighted transfer theorem: if every nonnegative integer weighting of a source-free digraph $D$ admits a quasi-kernel whose closed out-neighbourhood contains at least a $θ$-fraction of the total weight, then $D$ has a quasi-kernel of size at most $\\lfloor(1-θ)α(D)+θμ\\rfloor$ for $0<θ\\le1/2$, where $μ$ is the maximum matching size between a maximum independent set and its complement. Using the independent blow-up construction of Spiro, we deduce that the universal small and large quasi-kernel assertions are equivalent for every such $θ$, with no loss in the parameter. In particular, the two conjectures are equivalent. As an application, we prove the half-order bound for source-free break digraphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23487",
  "author_total": 4,
  "authors": [
    {
      "name": "Jiangdong Ai"
    },
    {
      "name": "Gregory Gutin"
    },
    {
      "name": "Xiangzhou Liu"
    },
    {
      "name": "Xianhao Meng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c768zqhhshbqq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23487",
  "fulltext_url": null,
  "id": "c768zqhhshbqq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23487",
  "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": "W7222939431",
  "pdf_url": "https://arxiv.org/pdf/2610.23487",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c768zqhhshbqq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.23487",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23487"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Small and large quasi-kernels: a weighted transfer theorem",
  "title_text": "Small and large quasi-kernels: a weighted transfer theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/c768zqhhshbqq",
  "venue": "arXiv",
  "versions": []
}