{
  "schema_version": 1,
  "problem_number": "OWR-13353-012",
  "title": "Computing the Odd-Ear Prefix Parameter Is NP-Complete",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a 2-vertex-connected graph G, let beta(G) be the maximum number of initial odd ears in an open ear decomposition, counting its initial cycle. We prove that deciding beta(G) >= k, with k part of the input, is NP-complete. This gives a complexity classification for a question of Benchetrit and Sebo. Hardness holds even for simple, 2-connected, 2-degenerate graphs whose deletion of one vertex is bipartite, and whose girth is at least any prescribed fixed bound. The proof identifies beta on a cone over a bipartite multigraph with the largest induced edge count on a connected vertex set admitting a perfect matching. Two polynomially bounded edge multiplicities adapt a known connected-matching satisfiability gadget to this different objective. Odd subdivision removes parallel edges without changing beta. The known gadget and theorem inputs are credited. This is an AI-assisted, self-audited, unrefereed preprint; no independent verification or absolute priority claim is made.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "cs.CC",
    "cs.DM"
  ],
  "keywords": [
    "OWR-13353-012",
    "odd ear decomposition",
    "factor-critical graph",
    "NP-completeness",
    "connected matching",
    "odd subdivision"
  ],
  "manuscript_version_date": "2026-10-23",
  "publication_date": "2026-10-23",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-23",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-13353-012/",
  "pdf_url": "https://eulersolve.org/papers/owr-13353-012/paper.pdf?v=91e18175e559",
  "doi": "10.5281/zenodo.23592264",
  "zenodo_record_url": "https://zenodo.org/records/23592264",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Deciding the maximum initial odd-ear count beta(G)>=k is NP-complete, even for simple 2-connected 2-degenerate apex-bipartite graphs of arbitrarily large fixed girth. This is a complexity classification, not a proof that P differs from NP. The preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "91e18175e559c2362ff357f29c23bed98971c5472b47a8b300187353e7755820"
    },
    "source.zip": {
      "sha256": "b1a2199064f9788f6977661aea370f014d1f42bdf67a5c3098d7877a2a281753"
    },
    "verification_report.md": {
      "sha256": "d916005566318108c36f766b4242487df9d30e37c5971caf420043c0dd92ab3a"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}