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We show that one of the following holds: $G-x_1$ contains $K_4^-$, or $G$ contains a $K_4^-$ in which $x_1$ is of degree 2, or $G$ contains a $TK_5$ in which $x_1$ is not a branch vertex, or $\\{x_2,y_1,y_2\\}$ may be chosen so that for any distinct $z_0, z_1\\in N(x_1)-\\{x_2,y_1,y_2\\}$, $G-\\{x_1v:v\\notin \\{z_0, z_1,x_2, y_1,y_2\\}\\}$ contains $TK_5$. This result will be used to prove the Kelmans-Seymour conjecture.","authors_text":"Dawei He, Xingxing Yu, Yan Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-19T14:40:15Z","title":"The Kelmans-Seymour conjecture III: 3-vertices in $K_4^-$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05747","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ed8e31f47d77d40a1756aefe60760dec48074dead82135c6ea5ddd8f0acbfd91","target":"record","created_at":"2026-05-18T01:04:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"eb85ece1e266f154433229817a2a1b0d3abbf5e9189d88685b7ca9eb6c40e1f7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-19T14:40:15Z","title_canon_sha256":"3a864124662c8d4fd9bb7856ef832be93caf40ad883225c7d9191841116be4bc"},"schema_version":"1.0","source":{"id":"1609.05747","kind":"arxiv","version":1}},"canonical_sha256":"8abc2b83610dd0de0017bdda9bb70ec7e20876983d4c057e51daea29471014e3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8abc2b83610dd0de0017bdda9bb70ec7e20876983d4c057e51daea29471014e3","first_computed_at":"2026-05-18T01:04:25.421108Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:25.421108Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9HoFxyMiH5GK+gWW35DWsu9/sVFnS7T2DPcg9oQ1EbgW9/Anx027n3YyTmxHa33QAOGQSd93ewK6qyCjgOTODQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:25.421753Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.05747","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ed8e31f47d77d40a1756aefe60760dec48074dead82135c6ea5ddd8f0acbfd91","sha256:2a869879607349a5868d6b2d828fbef3bd432a8665c913dc1be45589e37bee42"],"state_sha256":"ad881924b65301bbdee5a67ccbe5b34303a6f03b92e0aac324c6e58b572ffde4"}