A Polynomial Bound for Untangling Geometric Planar Graphs

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 . We also consider untangling geometric trees. It is known that every n -vertex geometric tree can be untangled while keeping at least vertices fixed, while the best upper bound was . 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 vertices fixed.