, Volume 42, Issue 4, pp 570-585

A Polynomial Bound for Untangling Geometric Planar Graphs

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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.

Research of P. Bose and P. Morin was partially supported by NSERC.
Research of V. Dujmović was partially supported by CRM and NSERC.
Research of F. Hurtado was supported by projects MEC MTM2006-01267 and DURSI 2005SGR00692.
Research of D.R. Wood was supported by a QEII Research Fellowship. Research initiated at the Universitat Politècnica de Catalunya, where supported by the Marie Curie Fellowship MEIF-CT-2006-023865, and by projects MEC MTM2006-01267 and DURSI 2005SGR00692.