{"paper":{"title":"Untangling planar graphs from a specified vertex position - Hard cases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"cs.DM","authors_text":"Alexander Ravsky, Mathias Schacht, Mihyun Kang, Oleg Pikhurko, Oleg Verbitsky","submitted_at":"2008-03-06T13:08:41Z","abstract_excerpt":"Given a planar graph $G$, we consider drawings of $G$ in the plane where edges are represented by straight line segments (which possibly intersect). Such a drawing is specified by an injective embedding $\\pi$ of the vertex set of $G$ into the plane. We prove that a wheel graph $W_n$ admits a drawing $\\pi$ such that, if one wants to eliminate edge crossings by shifting vertices to new positions in the plane, then at most $(2+o(1))\\sqrt n$ of all $n$ vertices can stay fixed. Moreover, such a drawing $\\pi$ exists even if it is presupposed that the vertices occupy any prescribed set of points in t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0803.0858","kind":"arxiv","version":5},"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"}