{
  "abstract": "A graph $G$ is $H$-saturated if $G$ contains no $H$-copy as a subgraph, but adding any edge between two non-adjacent vertices in $G$ creates a copy of $H$. The saturation number $\\mathrm{sat}(n,H)$ is the minimum number of edges in an $n$-vertex $H$-saturated graph. Cameron and Puleo [Discrete Math. 345 (2022), 112867] showed that $\\mathrm{sat}(n,K_1\\vee F)\\le n-1+\\mathrm{sat}(n-1,F)$ for all $n>|V(F)|$. It is natural to ask when the equality holds. Most known results on this bound consider non-empty graphs $F$ without isolated vertices. We study families in which $F$ contains isolated vertices. For $F=K_{p-1}\\cup qK_1$, where $p\\ge4$ and $q\\ge p-1$, we prove that $\\mathrm{sat}(n,K_1\\vee F)=(p-2)n-\\binom{p-1}{2}$ for $n\\ge\\max\\{p+q,\\binom p2\\}$ and determine the unique extremal graph. We also give the exact result for $q=1$, where the extremal graph has a different structure. For $F=pK_2\\cup qK_1$, in the natural range $n\\ge2p+q+1$, we prove that $\\mathrm{sat}(n,K_1\\vee F)=n+3p-4$ holds exactly when $n\\ge3p-2$, and the order condition is sharp. Furthermore, we characterize all extremal graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.22011",
  "author_total": 2,
  "authors": [
    {
      "name": "Xinying Hua"
    },
    {
      "name": "Yuejian Peng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6hmm142pdcm59.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.22011",
  "fulltext_url": null,
  "id": "6hmm142pdcm59",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.22011",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7165627625",
  "pdf_url": "https://arxiv.org/pdf/2606.22011",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6hmm142pdcm59.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2606.22011",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.22011"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Saturation numbers for joins with graphs containing isolated vertices",
  "title_text": "Saturation numbers for joins with graphs containing isolated vertices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/6hmm142pdcm59",
  "venue": "arXiv",
  "versions": []
}