{
  "abstract": "The (global) average order of dominating sets of a graph is the average number of vertices of its dominating sets. Analogously, the local average order of dominating sets is the average number of vertices of its dominating sets containing a fixed vertex. In this paper, we show that the local average order of dominating sets of a graph with $n$ vertices is at least $\\frac{n+1}{2}$, with equality if and only if the degree of the fixed vertex is $n-1$. Furthermore, for a graph on $n$ vertices without isolated vertices, we show that $\\frac{5n-1}{6}$ is an upper bound for the local average order of dominating sets. Additionally, we give a proof of an exact formula for the local average order of dominating sets when the degree of the fixed vertex is $n-2$, and determine an upper bound for the local average order of dominating sets when the fixed vertex is an $l$-stem ($l \\geq 2$).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11446",
  "author_total": 4,
  "authors": [
    {
      "name": "Tingyun Chen"
    },
    {
      "name": "Weihua He"
    },
    {
      "name": "Hong-Jian Lai"
    },
    {
      "name": "Jianping Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cevgandv1mhx1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11446",
  "fulltext_url": null,
  "id": "cevgandv1mhx1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11446",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7221086388",
  "pdf_url": "https://arxiv.org/pdf/2610.11446",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cevgandv1mhx1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11446",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11446"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the local average order of dominating sets",
  "title_text": "On the local average order of dominating sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/cevgandv1mhx1",
  "venue": "arXiv",
  "versions": []
}