{"paper":{"title":"Six variations on a theme: almost planar graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.CO","authors_text":"Eoin Mackall, Ilan Weinschelbaum, Jeremy Thomas, Max Lipton, Mike Pierce, Samantha Robinson, Thomas W. Mattman","submitted_at":"2016-08-05T19:00:00Z","abstract_excerpt":"A graph is apex if it can be made planar by deleting a vertex, that is, $\\exists v$ such that $G-v$ is planar. We define the related notions of edge apex, $\\exists e$ such that $G-e$ is planar, and contraction apex, $\\exists e$ such that $G/e$ is planar, as well as the analogues with a universal quantifier: $\\forall v$, $G-v$ planar; $\\forall e$, $G-e$ planar; and $\\forall e$, $G/e$ planar. The Graph Minor Theorem of Robertson and Seymour ensures that each of these six gives rise to a finite set of obstruction graphs. For the three definitions with universal quantifiers we determine this set. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01973","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"}