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Given a drawing of a graph, its \emph{visual complexity} is defined as thenumber of geometrical entities in the drawing, for example, the number ofsegments in a straight-line drawing or the number of arcs in a circular-arcdrawing (in 2D). Recently, Chaplick et al. [GD 2016] introduced a differentmeasure for the visual complexity, the \emph{affine cover number}, which is theminimum number of lines (or planes) that together cover a crossing-freestraight-line drawing of a graph $G$ in 2D (3D). In this paper, we introducethe \emph{spherical cover number}, which is the minimum number of circles (orspheres) that together cover a crossing-free circular-arc drawing in 2D (or3D). It turns out that spherical covers are sometimes significantly smallerthan affine covers. Moreover, there are highly symmetric graphs that havesymmetric optimum spherical covers but apparently no symmetric optimum affinecover. For complete, complete bipartite, and platonic graphs, we analyze theirspherical cover numbers and compare them to their affine cover numbers as wellas their segment and arc numbers. We also link the spherical cover number toother graph parameters such as chromatic number, treewidth, and lineararboricity.

Drawing Graphs on Few Circles and Few Spheres