Upper Bound Constructions for Untangling Planar Geometric Graphs

Abstract

For every $n\in \mathbb{N}$, we construct an $n$-vertex planar graph $G=(V,E)$ and $n$ distinct points $p(v)$, $v\in V$, in the plane such that in any crossing-free straight-line drawing of $G$, at most $O(n^{.4948})$ vertices $v\in V$ are embedded at points $p(v)$. This improves on an earlier bound of $O(\sqrt{n})$ by Goaoc et al. [Discrete Comput. Geom., 42 (2009), pp. 542--569].