Ordered Level Planarity, Geodesic Planarity and Bi-Monotonicity

TL;DR

This paper introduces and analyzes the computational complexity of Ordered Level Planarity, Geodesic Planarity, and Bi-Monotonicity problems, revealing new NP-hardness results and connections to existing graph drawing problems.

Contribution

It establishes a complexity dichotomy for Ordered Level Planarity and proves NP-hardness for Geodesic Planarity with multiple directions, extending to Bi-Monotonicity, and answers open questions in the field.

Findings

Ordered Level Planarity is NP-complete for certain parameters.

Geodesic Planarity with four or more directions is NP-hard.

The reduction to Clustered Level Planarity involves instances with only two non-trivial clusters.

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 is realized as a polygonal path composed of line segments with two adjacent directions from a given set $S$ of directions symmetric with respect to the origin. Our results on Ordered Level Planarity imply $NP$-hardness for any $S$ with $|S|\ge 4$ even if the given graph is a matching. Katz, Krug, Rutter and Wolff claimed that for matchings Manhattan Geodesic Planarity, the case where $S$ contains precisely the horizontal and vertical directions, can be solved in polynomial time [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 \v{S}tefankovi\v{c}. 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.