1-bend upward planar drawings of SP-digraphs

Abstract

It is proved that every series-parallel digraph whose maximum vertex-degree is Δ admits an upward planar drawing with at most one bend per edge such that each edge segment has one of Δ distinct slopes. This is shown to be worst-case optimal in terms of the number of slopes. Furthermore, our construction gives rise to drawings with optimal angular resolution π/Δ. A variant of the proof technique is used to show that (non-directed) reduced series-parallel graphs and flat series-parallel graphs have a (non-upward) one-bend planar drawing with ⎾Δ/2⏋ distinct slopes if biconnected, and with ⎾Δ/2⏋+1 distinct slopes if connected.