{
  "abstract": "For a positive integer $k$, let $β_k(G)$ be the maximum cardinality of a vertex set inducing maximum degree less than $k$, and let $SF_k(G)$ be the maximum cardinality of a minimal $k$-star-forming set. The known inequality $β_k(G)\\le SF_k(G)$ suggests asking when equality holds. We place this question in the framework of upper domination: at $k=1$, $β_1(G)=α(G)$ and $SF_1(G)=Γ(G)$, so the classical equality $Γ=α$ on bipartite graphs is exactly the first member of the proposed hierarchy. We prove the equality for complete bipartite graphs for every $k$, obtaining \\[ β_k(K_{a,b})=SF_k(K_{a,b})=\\max\\{a,b,2k-2\\}\\qquad(a,b\\ge k), \\] and record the elementary low-degree case $Δ(G)<k$. For $k=2$ we derive certificate restrictions for minimal $2$-star-forming sets in bipartite graphs. For chain graphs we go further: we prove $β_2(G)=SF_2(G)$ and obtain an exact formula for their common value. The proof uses the nested-neighborhood structure together with a classification of witnesses to the indispensability of a high internal-degree vertex. We retain the equality problem for chain graphs as a conjecture only for $k\\ge3$, and formulate the broader bipartite conjecture. We also determine the upper domination number of every rectangular grid and combine it with the known exact dissociation number to compare $Γ$, $β_2$, and $SF_2$. In particular, $Γ=β_2$ on every even-by-even rectangular grid, while $β_2\\le SF_2$ always; this motivates a grid equality conjecture whose even-by-even case would yield a three-parameter identity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03785",
  "author_total": 1,
  "authors": [
    {
      "name": "Rafik Sahbi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4j98j13s9ad5g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03785",
  "fulltext_url": null,
  "id": "4j98j13s9ad5g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03785",
  "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": "W7220723887",
  "pdf_url": "https://arxiv.org/pdf/2610.03785",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4j98j13s9ad5g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.03785",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03785"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Upper $k$-Star-Forming Sets, $k$-Independence, and Upper Domination",
  "title_text": "Upper k-Star-Forming Sets, k-Independence, and Upper Domination",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/4j98j13s9ad5g",
  "venue": "arXiv",
  "versions": []
}