Bitonic st-orderings for Upward Planar Graphs
  Canonical orderings serve as the basis for many incremental planar drawing algorithms. All these techniques, however, have in common that they are limited to undirected graphs. While $st$-orderings do extend to directed graphs, especially planar $st$-graphs, they do not offer the same properties as canonical orderings. In this work we extend the so called bitonic $st$-orderings to directed graphs. We fully characterize planar $st$-graphs that admit such an ordering and provide a linear-time algorithm for recognition and ordering. If for a graph no bitonic $st$-ordering exists, we show how to find in linear time a minimum set of edges to split such that the resulting graph admits one. With this new technique we are able to draw every upward planar graph on $n$ vertices by using at most one bend per edge, at most $n - 3$ bends in total and within quadratic area. 
Martin Gronemann
cs.DS