1-Bend Upward Planar Drawings of SP-Digraphs

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