{
  "abstract": "A set $S$ of vertices of a graph $G$ is a connected mutual-visibility set if every two vertices of $S$ are joined by a shortest path whose internal vertices lie outside $S$, and the subgraph induced by $S$ is connected. We introduce the connected mutual-visibility number $μ_c(G)$, defined as the maximum cardinality of such a set, and investigate its structural and algorithmic properties. We establish fundamental bounds, derive Nordhaus--Gaddum type inequalities, and characterise the graphs attaining the minimum and maximum possible values. For regular $(d,2,-δ)$-graphs, we derive general bounds on $μ_c(G)$ and determine its exact value for the two cubic graphs of defect $2$. We further show that $μ_c(G)$ is determined locally by the block structure of $G$, namely, it is equal to the maximum of the corresponding values over the blocks of $G$. Finally, we present a polynomial-time algorithm for recognising connected mutual-visibility sets and prove that the associated decision problem is $\\mathsf{NP}$-complete, even for connected bipartite graphs of diameter at most $4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18877",
  "author_total": 2,
  "authors": [
    {
      "name": "Tonny K B"
    },
    {
      "name": "Shikhi M"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e5bagzvjwsgw4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18877",
  "fulltext_url": null,
  "id": "e5bagzvjwsgw4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18877",
  "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": "W7213509023",
  "pdf_url": "https://arxiv.org/pdf/2609.18877",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e5bagzvjwsgw4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18877",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18877"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Connected Mutual-Visibility in Graphs",
  "title_text": "Connected Mutual-Visibility in Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/e5bagzvjwsgw4",
  "venue": "arXiv",
  "versions": []
}