{
  "abstract": "A mutual-visibility set in a graph $G$ is a set $S$ of vertices such that any two vertices of $S$ are connected by a shortest path with no internal vertex in $S$, and the mutual-visibility chromatic number $χ_μ(G)$ is the minimum number of mutual-visibility sets needed to partition $V(G)$. Babu, Di Stefano and Lakshmanan showed that it is NP-complete to decide whether $χ_μ(G)\\le 2$ for graphs of diameter four. We show that the problem remains NP-complete for graphs of diameter $d$, for every fixed $d\\ge2$; diameter two is the smallest diameter for which the problem is non-trivial. The proof rests on a reformulation of $χ_μ(G)\\le 2$ for diameter-two graphs as the $2$-colourability of an associated hypergraph, and on a reduction from $2$-colourability of $3$-uniform hypergraphs without short cycles.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01012",
  "author_total": 1,
  "authors": [
    {
      "name": "Siddhesh Chaubal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dvv3mwpks382w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01012",
  "fulltext_url": null,
  "id": "dvv3mwpks382w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01012",
  "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",
      "cs.CC"
    ],
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01012",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dvv3mwpks382w.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01012",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01012"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Mutual-visibility colouring with two colours is NP-complete for every fixed diameter",
  "title_text": "Mutual-visibility colouring with two colours is NP-complete for every fixed diameter",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:34Z",
  "url": "https://research.draconitic.com/paper/dvv3mwpks382w",
  "venue": "arXiv",
  "versions": []
}