{
  "abstract": "A dominating set $D$ of a graph is called $1$-limited if every vertex of $D$ has at most one neighbor outside $D$, while a $(1,2)$-dominating set is a dominating set in which every vertex of the set has at least two neighbors within the set. These two notions coincide on cubic graphs. We prove that the decision problem 1-Limited Dominating Set is $\\mathsf{NP}$-complete even when restricted to $2$-connected planar cubic graphs, thereby completing the known complexity results for $k$-Limited Dominating Set for all fixed positive integers $k$. We also determine the exact $1$-limited domination number of the entire Goldberg family. This provides a further infinite family of cubic graphs supporting several open conjectures and proposed bounds concerning $(1,2)$-domination and induced cycles.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01796",
  "author_total": 2,
  "authors": [
    {
      "name": "Goran Radić"
    },
    {
      "name": "Aleksandra Tepeh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/934g078x1v8n8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01796",
  "fulltext_url": null,
  "id": "934g078x1v8n8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01796",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.01796",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/934g078x1v8n8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01796",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01796"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On $1$-limited and $(1,2)$-domination in cubic graphs",
  "title_text": "On $1$-limited and (1,2)-domination in cubic graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/934g078x1v8n8",
  "venue": "arXiv",
  "versions": []
}