Drawing Graphs on Few Circles and Few Spheres
Myroslav Kryven, Alexander Ravsky, Alexander Wolff

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 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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
