Drawing Graphs on Few Circles and Few Spheres

TL;DR

This paper introduces the spherical cover number as a new measure of visual complexity for graph drawings, analyzing its properties and relationships with other graph parameters, and comparing it to the affine cover number.

Contribution

It defines the spherical cover number for circular-arc graph drawings and explores its advantages over affine covers, including symmetry and size considerations.

Findings

Spherical cover numbers can be smaller than affine cover numbers.

Highly symmetric graphs may have symmetric spherical covers but not affine covers.

Spherical cover number relates to chromatic number, treewidth, and linear arboricity.

Abstract

Given a drawing of a graph, its \emph{visual complexity} is defined as the number of geometrical entities in the drawing, for example, the number of segments in a straight-line drawing or the number of arcs in a circular-arc drawing (in 2D). Recently, Chaplick et al. [GD 2016] introduced a different measure for the visual complexity, the \emph{affine cover number}, which is the minimum number of lines (or planes) that together cover a crossing-free straight-line drawing of a graph $G$ in 2D (3D). In this paper, we introduce the \emph{spherical cover number}, which is the minimum number of circles (or spheres) that together cover a crossing-free circular-arc drawing in 2D (or 3D). It turns out that spherical covers are sometimes significantly smaller than affine covers. Moreover, there are highly symmetric graphs that have symmetric optimum spherical covers but apparently no symmetric optimum affine cover. For complete, complete bipartite, and platonic graphs, we analyze their spherical cover numbers and compare them to their affine cover numbers as well as their segment and arc numbers. We also link the spherical cover number to other graph parameters such as chromatic number, treewidth, and linear arboricity.