{"paper":{"title":"The Kelmans-Seymour conjecture II: 2-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":"2016-02-24T15:27:02Z","abstract_excerpt":"We use $K_4^-$ to denote the graph obtained from $K_4$ by removing an edge, and use $TK_5$ to denote a subdivision of $K_5$. Let $G$ be a 5-connected nonplanar graph and $\\{x_1,x_2,y_1,y_2\\}\\subseteq V(G)$ such that $G[\\{x_1,x_2,$ $y_1,y_2\\}]\\cong K_4^-$ with $y_1y_2\\notin E(G)$. Let $w_1,w_2,w_3\\in N(y_2)-\\{x_1,x_2\\}$ be distinct. We show that $G$ contains a $TK_5$ in which $y_2$ is not a branch vertex, or $G-y_2$ contains $K_4^-$, or $G$ has a special 5-separation, or $G-\\{y_2v:v\\notin \\{w_1,w_2,w_3,x_1,x_2\\}\\}$ contains $TK_5$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.07557","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"}