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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1609.05747","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-19T14:40:15Z","cross_cats_sorted":[],"title_canon_sha256":"3a864124662c8d4fd9bb7856ef832be93caf40ad883225c7d9191841116be4bc","abstract_canon_sha256":"eb85ece1e266f154433229817a2a1b0d3abbf5e9189d88685b7ca9eb6c40e1f7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:25.421753Z","signature_b64":"9HoFxyMiH5GK+gWW35DWsu9/sVFnS7T2DPcg9oQ1EbgW9/Anx027n3YyTmxHa33QAOGQSd93ewK6qyCjgOTODQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8abc2b83610dd0de0017bdda9bb70ec7e20876983d4c057e51daea29471014e3","last_reissued_at":"2026-05-18T01:04:25.421108Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:25.421108Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Kelmans-Seymour conjecture III: 3-vertices in $K_4^-$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dawei He, Xingxing Yu, Yan Wang","submitted_at":"2016-09-19T14:40:15Z","abstract_excerpt":"Let $G$ be a 5-connected nonplanar graph and let $x_1,x_2,y_1,y_2\\in V(G)$ be distinct, such that $G[\\{x_1,x_2,y_1,y_2\\}]\\cong K_4^-$ and $y_1y_2\\notin E(G)$. 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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05747","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1609.05747","created_at":"2026-05-18T01:04:25.421231+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.05747v1","created_at":"2026-05-18T01:04:25.421231+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.05747","created_at":"2026-05-18T01:04:25.421231+00:00"},{"alias_kind":"pith_short_12","alias_value":"RK6CXA3BBXIN","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_16","alias_value":"RK6CXA3BBXIN4AAX","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_8","alias_value":"RK6CXA3B","created_at":"2026-05-18T12:30:41.710351+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RK6CXA3BBXIN4AAXXXNJXNYOY7","json":"https://pith.science/pith/RK6CXA3BBXIN4AAXXXNJXNYOY7.json","graph_json":"https://pith.science/api/pith-number/RK6CXA3BBXIN4AAXXXNJXNYOY7/graph.json","events_json":"https://pith.science/api/pith-number/RK6CXA3BBXIN4AAXXXNJXNYOY7/events.json","paper":"https://pith.science/paper/RK6CXA3B"},"agent_actions":{"view_html":"https://pith.science/pith/RK6CXA3BBXIN4AAXXXNJXNYOY7","download_json":"https://pith.science/pith/RK6CXA3BBXIN4AAXXXNJXNYOY7.json","view_paper":"https://pith.science/paper/RK6CXA3B","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.05747&json=true","fetch_graph":"https://pith.science/api/pith-number/RK6CXA3BBXIN4AAXXXNJXNYOY7/graph.json","fetch_events":"https://pith.science/api/pith-number/RK6CXA3BBXIN4AAXXXNJXNYOY7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RK6CXA3BBXIN4AAXXXNJXNYOY7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RK6CXA3BBXIN4AAXXXNJXNYOY7/action/storage_attestation","attest_author":"https://pith.science/pith/RK6CXA3BBXIN4AAXXXNJXNYOY7/action/author_attestation","sign_citation":"https://pith.science/pith/RK6CXA3BBXIN4AAXXXNJXNYOY7/action/citation_signature","submit_replication":"https://pith.science/pith/RK6CXA3BBXIN4AAXXXNJXNYOY7/action/replication_record"}},"created_at":"2026-05-18T01:04:25.421231+00:00","updated_at":"2026-05-18T01:04:25.421231+00:00"}