{
  "abstract": "A dominating set $D$ of a graph $G$ is called $k$-limited if every vertex of $D$ has at most $k$ neighbors outside $D$. The minimum cardinality among all $k$-limited dominating sets of $G$ is the $k$-limited domination number, denoted by $γ_k^L(G)$. In this paper, we prove that the $1$-Limited Dominating Set problem is $\\mathsf{NP}$-complete, answering an open question posed in the literature. We further show that, for every positive integer $k$, the $k$-Limited Dominating Set problem is $\\mathsf{NP}$-complete even when restricted to planar graphs. In addition, we study the $k$-limited domination number of Sierpiński graphs. We determine the exact value of $γ_1^L(S(n,m))$ for all integers $n\\ge 1$ and $m\\ge 2$, and obtain the $k$-limited domination number for the planar family $S(n,3)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01584",
  "author_total": 1,
  "authors": [
    {
      "name": "Dragana Božović"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e3trep983cbfn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01584",
  "fulltext_url": null,
  "id": "e3trep983cbfn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01584",
  "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",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.01584",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e3trep983cbfn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01584",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01584"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On $k$-limited domination: complexity and Sierpiński graphs",
  "title_text": "On k-limited domination: complexity and Sierpiński graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/e3trep983cbfn",
  "venue": "arXiv",
  "versions": []
}