Untangling a Planar Graph
نویسندگان
چکیده
منابع مشابه
Untangling a Planar Graph
A straight-line drawing δ of a planar graph G need not be plane, but can be made so by untangling it, that is, by moving some of the vertices of G. Let shift(G, δ) denote the minimum number of vertices that need to be moved to untangle δ. We show that shift(G, δ) is NP-hard to compute and to approximate. Our hardness results extend to a version of 1BendPointSetEmbeddability, a well-known graph-...
متن کاملUntangling Planar Curves
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 Θ(n3/2) homotopy moves in the worst case. Our algorithm improves the best previous upper bound O(n2), which is already implicit in the classical work of Steinitz; th...
متن کامل2 Untangling Planar Curves
6 Any generic closed curve in the plane can be transformed into a simple closed curve by a finite 7 sequence of local transformations called homotopy moves. We prove that simplifying a planar 8 closed curve with n self-crossings requires Θ(n3/2) homotopy moves in the worst case. Our 9 algorithm improves the best previous upper bound O(n2), which is already implicit in the classical 10 work of S...
متن کاملA Polynomial Bound for Untangling Geometric Planar Graphs
To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every n-vertex geometric planar graph can be untangled while keeping at least n vertices fixed. We answer this question in the affirmative with ǫ = 1/4. The previous best known bound was Ω( √ logn/ log logn). We also con...
متن کاملUntangling planar graphs from a specified vertex position - Hard cases
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 π of the vertex set of G into the plane. Let fix(G, π) be the maximum integer k such that there exists a crossing-free redrawing π of G which keeps k vertices fixed, i.e., there exist k vertices v1,...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Discrete & Computational Geometry
سال: 2009
ISSN: 0179-5376,1432-0444
DOI: 10.1007/s00454-008-9130-6