A polynomial bound for untangling geometric planar graphs

TL;DR

This paper proves that every n-vertex geometric planar graph can be untangled while fixing at least n^{1/4} vertices, improving previous bounds, and also refines bounds for untangling geometric trees.

Contribution

It establishes a polynomial lower bound for untangling geometric planar graphs and closes the gap for the maximum fixed vertices in untangling geometric trees.

Findings

Every n-vertex geometric planar graph can be untangled fixing at least n^{1/4} vertices.

For infinitely many n, some geometric trees cannot be untangled fixing more than 3(n^{1/2}-1) vertices.

Improved lower bound to (n/2)^{1/2} for untangling geometric trees.

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., 2002] asked if every n-vertex geometric planar graph can be untangled while keeping at least n^\epsilon vertices fixed. We answer this question in the affirmative with \epsilon=1/4. The previous best known bound was \Omega((\log n / \log\log n)^{1/2}). We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least (n/3)^{1/2} vertices fixed, while the best upper bound was O(n\log n)^{2/3}. We answer a question of Spillner and Wolff [arXiv:0709.0170 2007] 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(n^{1/2}-1) vertices fixed. Moreover, we improve the lower bound to (n/2)^{1/2}.