{
  "abstract": "For a graph $G$ without isolated vertices, $γ(G)\\leγ_t(G)\\le 2γ(G)$. While graphs attaining $γ(G)=γ_t(G)$ have been studied extensively, a complete structural description in the smallest nontrivial case $γ(G)=2$ has remained open. We resolve this case according to girth. When $g(G)\\ne 3$, we show $γ_t(G)=2$ forces $G$ bipartite, give an exact degree-sum criterion for this equality, and show $γ_t(G)\\in\\{2,4\\}$ under the additional hypothesis $δ(G)\\ge 2$. When $g(G)=3$, we use Golumbic's vertex-multiplication operation together with known classifications of graphs of rank $2$ through $5$ to completely list the families satisfying $γ(G)=γ_t(G)=2$. As an application, we show that every graph in the extremal family of diameter-two, dominating-vertex-free graphs identified by Erdős and Rényi and classified by Henning and Southey satisfies $γ_t(G)\\in\\{3,6\\}$, so $γ=γ_t=2$ never occurs there. Together these results give a full structural dictionary translating $γ_t(G)=2$ into concrete, checkable graph-theoretic properties.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03512",
  "author_total": 1,
  "authors": [
    {
      "name": "Sudip Bera"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fcjve07nwghts.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03512",
  "fulltext_url": null,
  "id": "fcjve07nwghts",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03512",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7208778826",
  "pdf_url": "https://arxiv.org/pdf/2609.03512",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fcjve07nwghts.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03512",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03512"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On graphs with equal domination and total domination numbers",
  "title_text": "On graphs with equal domination and total domination numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/fcjve07nwghts",
  "venue": "arXiv",
  "versions": []
}