Drawing Planar Graphs and 1-Planar Graphs Using Cubic B\'ezier Curves with Bounded Curvature

TL;DR

This paper presents algorithms for drawing planar and 1-planar graphs using cubic Bézier curves with bounded curvature, achieving efficient computation and specific aesthetic properties.

Contribution

It introduces algorithms for drawing planar and 1-planar graphs with cubic Bézier curves, ensuring bounded curvature and efficient computation within specified aesthetic constraints.

Findings

1-planar graphs can be drawn with a single cubic Bézier curve per edge in linear time.

Planar graphs can be drawn with bounded curvature and angular resolution in quadratic bounding boxes.

The methods are computationally efficient and produce aesthetically constrained graph drawings.

Abstract

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