Article

Discrete & Computational Geometry

, 42:570

First online:

A Polynomial Bound for Untangling Geometric Planar Graphs

  • Prosenjit BoseAffiliated withSchool of Computer Science, Carleton University Email author 
  • , Vida DujmovićAffiliated withDepartment of Mathematics and Statistics, McGill University
  • , Ferran HurtadoAffiliated withDepartament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya
  • , Stefan LangermanAffiliated withFNRS, Département d’Informatique, Université Libre de Bruxelles
  • , Pat MorinAffiliated withSchool of Computer Science, Carleton University
  • , David R. WoodAffiliated withDepartment of Mathematics and Statistics, The University of Melbourne

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Abstract

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. 28(4): 585–592, 2002) asked if every n-vertex geometric planar graph can be untangled while keeping at least n ε vertices fixed. We answer this question in the affirmative with ε=1/4. The previous best known bound was \(\Omega(\sqrt{\log n/\log\log n})\) . We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least \(\Omega(\sqrt{n})\) vertices fixed, while the best upper bound was \(\mathcal{O}((n\log n)^{2/3})\) . We answer a question of Spillner and Wolff (http://​arxiv.​org/​abs/​0709.​0170) by closing this gap for untangling trees. In particular, we show that for infinitely many values of n, there is an n-vertex geometric tree that cannot be untangled while keeping more than \(3(\sqrt{n}-1)\) vertices fixed.

Keywords

Geometric graphs Untangling Crossings