{
  "abstract": "The Roman Domination Problem (RDP) on a graph \\(G=(V,E)\\) asks for a labeling function \\(f:V\\rightarrow\\{0,1,2\\}\\) such that every vertex assigned value \\(0\\) is adjacent to a vertex assigned value \\(2\\). The objective is to minimize the total weight \\(\\sum_{v\\in V} f(v)\\); this minimum value is the Roman domination number of \\(G\\), denoted by \\(γ_R(G)\\). In this paper, we study RDP on graph classes motivated by convexity and induced-\\(P_4\\) structure. First, we consider circular-convex bipartite graphs, a natural superclass of convex bipartite graphs, where RDP is already known to be polynomial-time solvable. Assuming that a circular-convex representation is given, we compute \\(γ_R(G)\\) in \\(O(n^6)\\) time by cutting the circular order, separating interval and wrap-around vertices, and branching over at most two wrap-around vertices assigned value \\(2\\). Second, we study triad-convex bipartite graphs, a restricted subclass of tree-convex bipartite graphs whose convexity tree is a subdivision of \\(K_{1,3}\\). Although RDP is hard on broader tree-convex subclasses such as star-convex and comb-convex bipartite graphs, we show that \\(γ_R(G)\\) can be computed in \\(O(n^7)\\) time on triad-convex bipartite graphs. Finally, we study \\(P_4\\)-tidy graphs, which properly extend cographs. Using the Giakoumakis et al. structural decomposition of \\(P_4\\)-tidy graphs, we give a direct, exact algorithm that computes \\(γ_R(G)\\) in \\(O(n+m)\\) time. These results extend the algorithmic boundary of Roman domination on convexity-based bipartite graphs and \\(P_4\\)-structured graph classes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19240",
  "author_total": 2,
  "authors": [
    {
      "name": "Gautam K. Das"
    },
    {
      "name": "Kamal Santra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/csfnwh9vtc31d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19240",
  "fulltext_url": null,
  "id": "csfnwh9vtc31d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19240",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.19240",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/csfnwh9vtc31d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.19240",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19240"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Roman Domination on Circular-Convex, Triad-Convex Bipartite Graphs and $P_4$-Tidy Graphs",
  "title_text": "Roman Domination on Circular-Convex, Triad-Convex Bipartite Graphs and P_4-Tidy Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/csfnwh9vtc31d",
  "venue": "arXiv",
  "versions": []
}