{"paper":{"title":"A Kuratowski-Type Theorem for Planarity of Partially Embedded Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Ignaz Rutter, Jan Kratochv\\'il, V\\'it Jel\\'inek","submitted_at":"2012-04-13T08:28:11Z","abstract_excerpt":"A partially embedded graph (or PEG) is a triple (G,H,\\H), where G is a graph, H is a subgraph of G, and \\H is a planar embedding of H. We say that a PEG (G,H,\\H) is planar if the graph G has a planar embedding that extends the embedding \\H.\n  We introduce a containment relation of PEGs analogous to graph minor containment, and characterize the minimal non-planar PEGs with respect to this relation. We show that all the minimal non-planar PEGs except for finitely many belong to a single easily recognizable and explicitly described infinite family. We also describe a more complicated containment "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.2915","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"}