{
  "abstract": "We consider a natural extension of the classical saturation problem to oriented graphs, defined as the minimum number of edges in an oriented graph $H$ on $n$ vertices that avoids a fixed oriented graph $G$, but adding any missing edge to $H$ creates a copy of $G$. Our main result shows that saturation functions of oriented graphs attain at least four distinct orders of magnitude. In particular, if $G$ is acyclic, then its saturation function is bounded if and only if $G$ contains an isolated edge; otherwise, it is linear. Certain tournaments together with an isolated edge give further infinite examples of bounded and linear saturation functions. Two additional orders of magnitude are $Θ(n \\sqrt{\\log n})$ and $Θ(n \\log n)$. For these two, the lower bounds are obtained by adapting a recently developed approach for the saturation problem in edge-ordered graphs, while the matching upper bounds are constructive. Finally, we study the semisaturation problem, in which we require only that adding any missing edge to $H$ creates a copy of $G$. We completely characterize the possible orders of magnitude: the semisaturation function is either bounded, linear, or $Θ(n \\log n)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18600",
  "author_total": 2,
  "authors": [
    {
      "name": "Vladimir Bošković"
    },
    {
      "name": "Balázs Keszegh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9abd30h9444ze.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18600",
  "fulltext_url": null,
  "id": "9abd30h9444ze",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18600",
  "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": "W7221790629",
  "pdf_url": "https://arxiv.org/pdf/2610.18600",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9abd30h9444ze.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18600",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18600"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Saturation and semisaturation of oriented graphs",
  "title_text": "Saturation and semisaturation of oriented graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/9abd30h9444ze",
  "venue": "arXiv",
  "versions": []
}