Given a drawing of a graph, its 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.  introduced a different measure for the visual complexity, the affine cover number , which is the minimum number of lines (or planes) that together cover a  straight-line drawing of a graph G in 2D (3D). In this paper, we introduce the spherical cover number , which is the  number of circles (or spheres) that together cover a  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.
Given a drawing of a graph, its 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 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 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.
R
visual complexity
visual complexity
A
GD 2016
R
affine cover number
affine cover number
A
crossing-free
R
spherical cover number
spherical cover number
A
minimum
A
crossing-free