Improving the running time of embedded upward planarity testing

We consider the standard algorithm by Bertolazzi et al. to test the upward planarity of embedded digraphs. We show how to improve its running time from O(n+r²) to , where r is the number of sources and sinks in the digraph. We also discuss an application of this technique: improving the running time of getting a quasi-upward planar drawing for an embedded digraph with minimum number of bends.