Ordered Level Planarity and Its Relationship to Geodesic Planarity, Bi-Monotonicity, and Variations of Level Planarity

Abstract

We introduce and study the problem Ordered Level Planarity, which asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a y -monotone curve. This can be interpreted as a variant of Level Planarity in which the vertices on each level appear in a prescribed total order. We establish a complexity dichotomy with respect to both the maximum degree and the level-width, that is, the maximum number of vertices that share a level. Our study of Ordered Level Planarity is motivated by connections to several other graph drawing problems. Geodesic Planarity asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge e is realized as a polygonal path p composed of line segments with two adjacent directions from a given set S of directions that is symmetric with respect to the origin. Our results on Ordered Level Planarity imply NP -hardness for any S with ∣S∣ ≥ 4, even if the given graph is a matching. Manhattan Geodesic Planarity is the special case where S contains precisely the horizontal and vertical directions. Katz, Krug, Rutter, and Wolff claimed that Manhattan Geodesic Planarity can be solved in polynomial time for the special case of matchings [GD’09]. Our results imply that this is incorrect unless P = NP . Our reduction extends to settle the complexity of the Bi-Monotonicity problem, which was proposed by Fulek, Pelsmajer, Schaefer, and Štefankovič. Ordered Level Planarity turns out to be a special case of T-Level Planarity, Clustered Level Planarity, and Constrained Level Planarity. Thus, our results strengthen previous hardness results. In particular, our reduction to Clustered Level Planarity generates instances with only two non-trivial clusters. This answers a question posed by Angelini, Da Lozzo, Di Battista, Frati, and Roselli.