Upward planar drawings on the standing and the rolling cylinders

We consider directed graphs with an upward planar drawing on the standing and the rolling cylinders. These graphs properly extend the upward planar graphs in the plane. The drawings allow complex curves for the edges with many zig-zags and windings around the cylinder.We show that the drawings can be simplified to polyline drawings with geodesics as straight segments, at most two bends per edge, and vertices and bends on a grid of at most cubic size. On the standing cylinder the edges do not wind and they wind at most once around the rolling cylinder, where single windings can be unavoidable. The simplifications can be computed efficiently in O(τn3)O(τn3) time, where τ is the cost of computing the point of intersection of a curve and a horizontal line through a vertex.Moreover, we provide a complete classification of the classes of (regular) upward planar drawable graphs on the plane, the sphere, and the standing and rolling cylinders including their strict and weak variants. Strict is upward in two dimensions and weak allows horizontal lines. For upward planarity the standing and the rolling cylinders coincide in the strict case, rolling dominates standing in the regular case, and there is an incomparability in the weak case. The respective classes of graphs range properly between the upward planar and the quasi-upward planar graphs, and they have an NPNP-hard recognition problem.