文章基本信息

标题：Untangling Planar Curves
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作者：Hsien-Chih Chang ; Jeff Erickson
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2016
卷号：51
页码：29:1-29:16
DOI：10.4230/LIPIcs.SoCG.2016.29
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with n self-crossings requires Theta(n^{3/2}) homotopy moves in the worst case. Our algorithm improves the best previous upper bound O(n^2), which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold. This lower bound also implies that Omega(n^{3/2}) degree-1 reductions, series-parallel reductions, and Delta-Y transformations are required to reduce any planar graph with treewidth Omega(sqrt{n}) to a single edge, matching known upper bounds for rectangular and cylindrical grid graphs. Finally, we prove that Omega(n^2) homotopy moves are required in the worst case to transform one non-contractible closed curve on the torus to another; this lower bound is tight if the curve is homotopic to a simple closed curve.
关键词：computational topology; homotopy; planar graphs; Delta-Y transformations; defect; Reidemeister moves; tangles