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; ( i i ) it can be solved in linear time when an upward planar embedding of G is fixed; ( i i i ) 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; ( i v ) 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. • We study the algorithmic complexity of a problem that combines two well-established topics in graph drawing, namely rectilinear planar drawings and upward planar drawings. This problems, called rectilinear-upward planarity testing, asks to decide whether an input planar di-graph admits a planar drawing where each edge is either a horizontal or a vertical segment, and no edge points downwards. • We prove that rectilinear-upward planarity testing is NP-complete, even for biconnected digraphs. • We provide a linear-time algorithm for rectilinear-upward planarity testing of digraphs with a fixed upward planar embedding. • We provide a quadratic-time algorithm for rectilinear-upward planarity testing of biconnected partial 2-trees (i.e., digraphs whose underlying undirected graph is series-parallel) in the variable embedding setting. • We provide a fixed-parameter linear (FPL) algorithm for rectilinear- upward planarity testing of general biconnected digraphs in the variable embedding setting.