Drawing Planar Graphs and 1-Planar Graphs Using Cubic Bézier Curves with Bounded Curvature

TL;DR

It is shown that any n-vertex 1-planar graph has a 1-planar RAC drawing using a single cubic B\'ezier curve per edge, and this drawing can be computed in O(n) time given a combinatorial 1-planar drawing.

Abstract

We study algorithms for drawing planar graphs and 1-planar graphs using cubic Bézier curves with bounded curvature. We show that any n-vertex 1-planar graph has a 1-planar RAC drawing using a single cubic Bézier curve per edge, and this drawing can be computed in $O(n)$ time given a combinatorial 1-planar drawing. We also show that any n-vertex planar graph G can be drawn in $O(n)$ time with a single cubic Bézier curve per edge, in an $O(n)\times O(n)$ bounding box, such that the edges have $Θ(1/degree(v))$ angular resolution, for each $v \in G$, and $O(\sqrt{n})$ curvature.