A polynomial bound for untangling geometric planar graphs

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 $n^\in{}$ vertices fixed. We answer this question in the affirmative with ∊ = 1/4. The previous best known bound was Ω$(\sqrt{log\,n/log\,log\,n})$. We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least $(\sqrt{n/3})$ vertices fixed, while the best upper bound was 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.