{
  "abstract": "In a digraph $D$, a set $Q$ of vertices is a quasi-kernel if $Q$ is an independent set and every vertex in $D$ can be reached from $Q$ via at most two edges. The small quasi-kernel conjecture by Erdős and Székely predicts that every source-free $n$-vertex digraph contains a quasi-kernel of size at most $n - n/2$. Chvátal and Lovász' 1974 work implies a bound of $n - \\sqrt{n}+1$. In 2024, Sprio shows the bound $n - \\frac{1}{4} \\sqrt{n \\log_2 n}$. We show that there always is a quasi-kernel of size at most $n - \\frac{1}{51}n^{2/3} / \\ln^{2/3}(n)$. For the related large quasi-kernel conjecture by Spiro, we show that there always is a quasi-kernel $Q$ with $|N^+[Q]| \\ge 0.2917 \\cdot n^{2/3} / \\ln^{2/3}(n)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29659",
  "author_total": 1,
  "authors": [
    {
      "name": "Daniel Rutschmann"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ffrcc88p794gk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29659",
  "fulltext_url": null,
  "id": "ffrcc88p794gk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29659",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7223241125",
  "pdf_url": "https://arxiv.org/pdf/2610.29659",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ffrcc88p794gk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29659",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29659"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Small and Large Quasi-Kernels in General Digraphs",
  "title_text": "Small and Large Quasi-Kernels in General Digraphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/ffrcc88p794gk",
  "venue": "arXiv",
  "versions": []
}