Bitonic st-orderings for Upward Planar Graphs

Abstract: 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.