Quasi-Upward Planarity

all the edges are represented by curves monotonically increasing in
the vertical direction. We introduce the quasi-upward planar drawing convention,
extending the upward planarity theory. We give a polynomial time algorithm
for computing quasi-upward planar drawings with the minimum number of bends
of an embedded planar digraph.
We also provide a branch and bound algorithm for minimizing bends in
quasi-upward drawings, considering all the possible planar embeddings of the digraph