{"paper":{"title":"Non-planar extensions of subdivisions of planar graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Robin Thomas, Sergey Norin","submitted_at":"2014-02-09T22:29:34Z","abstract_excerpt":"Almost $4$-connectivity is a weakening of $4$-connectivity which allows for vertices of degree three. In this paper we prove the following theorem. Let $G$ be an almost $4$-connected triangle-free planar graph, and let $H$ be an almost $4$-connected non-planar graph such that $H$ has a subgraph isomorphic to a subdivision of $G$. Then there exists a graph $G'$ such that $G'$ is isomorphic to a minor of $H$, and either\n  (i) $G'=G+uv$ for some vertices $u,v\\in V(G)$ such that no facial cycle of $G$ contains both $u$ and $v$, or\n  (ii) $G'=G+u_1v_1+u_2v_2$ for some distinct vertices $u_1,u_2,v_1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.1999","kind":"arxiv","version":3},"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"}