Discrete & Computational Geometry

, 42:570

A Polynomial Bound for Untangling Geometric Planar Graphs

Authors

    • School of Computer ScienceCarleton University
  • Vida Dujmović
    • Department of Mathematics and StatisticsMcGill University
  • Ferran Hurtado
    • Departament de Matemàtica Aplicada IIUniversitat Politècnica de Catalunya
  • Stefan Langerman
    • FNRS, Département d’InformatiqueUniversité Libre de Bruxelles
  • Pat Morin
    • School of Computer ScienceCarleton University
  • David R. Wood
    • Department of Mathematics and StatisticsThe University of Melbourne
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