{"paper":{"title":"An excluded minor theorem for the 6-wheel","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"This paper classifies every 3-connected nonplanar graph without a 6-wheel minor, completing the full characterization of W6-minor-free graphs.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Weihua Yang, Yuqi Xu, Zijun Chen","submitted_at":"2026-05-14T17:37:15Z","abstract_excerpt":"For each integer $n \\geq 3$, the wheel graph $W_n$ is defined as the graph obtained by connecting a single vertex to all vertices of a cycle of length $n$. In particular, $W_6$ can be uniquely obtained from the Petersen graph by contracting three edges incident to a common vertex. Gubser provided a characterization of all 3-connected planar $W_6$-minor-free graphs. In this paper, we complete the characterization of $W_6$-minor-free graphs by determining the 3-connected nonplanar cases."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"In this paper, we complete the characterization of W_6-minor-free graphs by determining the 3-connected nonplanar cases.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Gubser's characterization of the planar 3-connected W_6-minor-free graphs is complete and correct, allowing the nonplanar cases to be exhaustively determined by the same minor-exclusion framework.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"All 3-connected nonplanar W_6-minor-free graphs are characterized.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"This paper classifies every 3-connected nonplanar graph without a 6-wheel minor, completing the full characterization of W6-minor-free graphs.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"eaf8c2099b06bb310378566d8d66f13aa81e463ded2e4a918833ee646d5d1d3f"},"source":{"id":"2605.15125","kind":"arxiv","version":1},"verdict":{"id":"7ae210e7-2324-4215-a991-73ceab65421e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T03:08:48.771273Z","strongest_claim":"In this paper, we complete the characterization of W_6-minor-free graphs by determining the 3-connected nonplanar cases.","one_line_summary":"All 3-connected nonplanar W_6-minor-free graphs are characterized.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Gubser's characterization of the planar 3-connected W_6-minor-free graphs is complete and correct, allowing the nonplanar cases to be exhaustively determined by the same minor-exclusion framework.","pith_extraction_headline":"This paper classifies every 3-connected nonplanar graph without a 6-wheel minor, completing the full characterization of W6-minor-free graphs."},"references":{"count":14,"sample":[{"doi":"","year":2007,"title":"G. Brinkmann and B. McKay. Fast generation of planar graphs.Match- Communications in Mathematical and in Computer Chemistry, 58(2):323–357, 2007. ISSN 0340-6253","work_id":"2502d32c-30fb-4ea8-9a63-5b81a98d48ce","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2013,"title":"G. Ding. A characterization of graphs with no octahedron minor.Journal of Graph Theory, 74(2):143–162, 2013","work_id":"5b9aa960-77e0-4398-9041-86e76749ef89","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2013,"title":"G. Ding and C. Liu. Excluding a small minor.Discrete Applied Mathematics, 161 (3):355–368, 2013","work_id":"6d4b1372-dcc3-4ed0-8626-b22a7d61ae50","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1952,"title":"G.A. Dirac. A property of 4-chromatic graphs and some remarks on critical graphs. Journal of the London Mathematical Society, 1(1):85–92, 1952","work_id":"6c413e98-19f8-4141-b431-0e9dd130f714","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"A.B. Ferguson. Excluding two minors of the petersen graph.Louisiana State Uni- versity and Agricultural and Mechanical College, 2015","work_id":"c2251b54-54e8-4748-aae2-50c741b69d11","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":14,"snapshot_sha256":"eab5a4ae2bc326e3fab97a2a5c3f4457d3d9d3192238966f99ea90a5a32ae9c8","internal_anchors":0},"formal_canon":{"evidence_count":1,"snapshot_sha256":"a6b93e11be86c3fc663586b858856cd483a04799676e05a05094fcdbf0859c2c"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}