{
  "schema_version": 1,
  "problem_number": "OWR-14299905-027",
  "title": "Independent-Root Counts Do Not Characterize Local Graph Removal",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give counterexamples to the fixed-cardinality independent-root criterion for local graph removal in Conjecture 8 of the 2026 Oberwolfach report Combinatorics. For any core graph with a maximal independent set smaller than its independence number, retaining that set as singleton parts and blowing up the other vertices gives graphs a positive normalized maximum-degree distance from being H-free, while every independent root count of the prescribed size is asymptotically small. A six-vertex wheel is an explicit example. For C9, the construction has 6m+3 vertices, maximum independent-four-root count m^4, and exact local removal distance ceil(m/2)/(6m+3), tending to 1/12. No single root cardinality can characterize local removal for such cores. The result concerns the literal printed criterion, including independence in the copy; it does not exclude formulations testing several root sizes or resolve the broader frozen UnsolvedMath OWR-14299905-027 question about all possible variants. The unpublished intended graph-removal manuscript was not obtained, so no claim about a corrected intended theorem is made. This five-page unrefereed preprint was developed with AI assistance and self-audited. Integer regression checks, independent small cycle enumeration and finite SAT deletion checks supplement the complete written proof; they are not independent referee review or formal proof-assistant verification.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "OWR-14299905-027",
    "local graph removal",
    "independent roots",
    "graph cores",
    "cycle blow-ups"
  ],
  "manuscript_version_date": "2026-10-27",
  "publication_date": "2026-10-27",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-27",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14299905-027/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299905-027/paper.pdf?v=06d207b5ed77",
  "doi": "10.5281/zenodo.23682808",
  "zenodo_record_url": "https://zenodo.org/records/23682808",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete counterexample to the exact fixed-cardinality independent-root criterion in Conjecture 8 of OWR1/2026 pp60-61. Non-well-covered cores yield a general obstruction; a six-vertex wheel and C9 are explicit. For C9, n=6m+3, maximum independent-four-root count m^4 and exact local removal distance ceil(m/2)/(6m+3). No single root size characterizes local removal for these cores. This refutes the literal printed Conjecture 8 of OWR1/2026. The broad frozen question about other local removal variants remains open; the unpublished intended corrected formulation was not obtained. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "06d207b5ed77c01e47a4fc7566d220297a2c0b181778f4c2eb2496788887f317"
    },
    "source.zip": {
      "sha256": "b854857e64c4a26fb78231284647494771946e8a933c3ef7ff42cd1e483839d3"
    },
    "verification_report.md": {
      "sha256": "6fea81c8e603883f47da8c21d0ca173450955179f401ec1db8c3ffba763e80ec"
    }
  },
  "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."
}