1-bend Upward Planar Drawings of SP-digraphs

Abstract

It is proved that every series-parallel digraph whose maximum vertex-degree is $\Delta$ admits an upward planar drawing with at most one bend per edge such that each edge segment has one of $\Delta$ 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 $\frac{\pi }{\Delta }$. 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 $\lceil \frac{\Delta }{2}\rceil$ distinct slopes if biconnected, and with $\lceil \frac{\Delta }{2}\rceil +1$ distinct slopes if connected.

Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016)