Rectilinear-upward planarity testing of digraphs

A rectilinear-upward planar drawing of a digraph G is a crossing-free drawing of G where each edge is either a horizontal or a vertical segment, and such that no directed edge points downward. RECTILINEAR-UPWARD PLANARITY TESTING is the problem of deciding whether a digraph G admits a rectilinear-upward planar drawing. We study the complexity of RECTILINEAR-UPWARD PLANARITY TESTING and provide several algorithmic results. Precisely, we prove that: (i) the problem is NP-complete, even if G is biconnected; (ii) it can be solved in linear time when an upward planar embedding of G is fixed; (iii) the problem is polynomial-time solvable for biconnected digraphs of treewidth at most two, i.e., for digraphs whose underlying undirected graph is a series-parallel graph; (iv) the problem is fixed-parameter tractable (namely, fixed-parameter linear) for all biconnected graphs, when parameterized by the number of sources and sinks in the digraph.