Discrete & Computational Geometry

, 42:570

A Polynomial Bound for Untangling Geometric Planar Graphs

  • Prosenjit Bose
  • Vida Dujmović
  • Ferran Hurtado
  • Stefan Langerman
  • Pat Morin
  • David R. Wood
Article

DOI: 10.1007/s00454-008-9125-3

Cite this article as:
Bose, P., Dujmović, V., Hurtado, F. et al. Discrete Comput Geom (2009) 42: 570. doi:10.1007/s00454-008-9125-3

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 nε 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 graphsUntanglingCrossings

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Prosenjit Bose
    • 1
  • Vida Dujmović
    • 2
  • Ferran Hurtado
    • 3
  • Stefan Langerman
    • 4
  • Pat Morin
    • 1
  • David R. Wood
    • 5
  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  3. 3.Departament de Matemàtica Aplicada IIUniversitat Politècnica de CatalunyaBarcelonaSpain
  4. 4.FNRS, Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
  5. 5.Department of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia