{
  "abstract": "A vertex is uni-dominated by a set S ⊆ V (G) if exactly one vertex of S lies in its closed neighborhood, and the uni-domination number ud * (G) is the largest number of vertices a single set can uni-dominate. The graphs admitting an efficient dominating set are those with ud * (G) = n. Hedetniemi and Rall asked for the largest constant c with ud * (T ) ≥ c • n for every tree T of order n. We answer their question: the constant is 4/5, and a set attaining it can be computed in linear time. This set is moreover a dominating set. Recall that an efficient dominating set satisfies two requirements at once: it dominates every vertex, and it dominates no vertex twice. Our set satisfies the first one, which leads us to study ud * under each of the two requirements separately, giving the Full-Cover variant ud * 0 and the Non-Overlapping variant ud * + . The two behave very differently: on arbitrarily large graphs ud * 0 may remain bounded by a constant, whereas ud * + always grows with the order. We present bounds for both, and show that the associated decision problems are polynomial for constant thresholds and NP-complete for the thresholds ⌈ c √ n⌉ and 4 √ n, respectively, the latter even on split graphs. Finally, we show that neither of the two parameters bounds the other.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 4,
  "authors": [
    {
      "name": "Florent Foucaud"
    },
    {
      "name": "Yan Gerard"
    },
    {
      "name": "Christian Laforest"
    },
    {
      "name": "Le Vu Pham"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a73neepkskf7r.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "a73neepkskf7r",
  "issns": [],
  "landing_url": "https://hal.science/hal-05768708v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.info"
    ],
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05768708/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a73neepkskf7r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:43Z",
      "native_id": "hal-05768708",
      "source": "hal",
      "url": "https://hal.science/hal-05768708v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Full-Cover and Non-Overlapping Uni-domination in Graphs: Bounds and Complexity",
  "title_text": "Full-Cover and Non-Overlapping Uni-domination in Graphs: Bounds and Complexity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:43Z",
  "url": "https://research.draconitic.com/paper/a73neepkskf7r",
  "venue": "HAL",
  "versions": []
}