{
  "abstract": "We prove that, on finite graphs of rank-width at most two, VR-recognizability and counting monadic second-order definability coincide. This advances the recognizability-versus-definability problem from bounded linear clique-width to the first nontrivial bounded rank-width level beyond the rank-width-one split-decomposition case. The proof first treats split-prime graphs. The maximal partial-tree theory of Clark and Whittle organizes the non-sequential cut-rank-two separations, while a single strong separation orients all strong equivalence classes and yields a CMSO-definable laminar family of canonical cores. Although the auxiliary partial tree is not itself transduced, it proves that every canonical local piece has a port-contiguous layout of uniformly bounded linear rank-width. The width argument uses partition atoms and the branch-width-three display theorem of Hall, Oxley, Semple, and Whittle and does not assume that graph torsos remain prime. Coherent ordered rank-two frames then permit a finite-state bottom-up evaluation whose local transitions are definable by the bounded-linear-clique-width theorem of Bojańczyk, Grohe, and Pilipczuk. Finally, the CMSO-transducible canonical split decomposition lifts the result from prime graphs to arbitrary graphs of rank-width at most two.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.10594",
  "author_total": 1,
  "authors": [
    {
      "name": "Antonios Kalampakas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1khy9ag169rxy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.10594",
  "fulltext_url": null,
  "id": "1khy9ag169rxy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.10594",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7168262877",
  "pdf_url": "https://arxiv.org/pdf/2607.10594",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1khy9ag169rxy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2607.10594",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.10594"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Recognizability equals CMSO-definability for graphs of rank-width at most two",
  "title_text": "Recognizability equals CMSO-definability for graphs of rank-width at most two",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/1khy9ag169rxy",
  "venue": "arXiv",
  "versions": []
}