Beyond Level Planarity

Abstract

In this paper we settle the computational complexity of two open problems related to the extension of the notion of level planarity to surfaces different from the plane. Namely, we show that the problems of testing the existence of a level embedding of a level graph on the surface of the rolling cylinder or on the surface of the torus, respectively known by the name of Cyclic Level Planarity and Torus Level Planarity, are polynomial-time solvable. Moreover, we show a complexity dichotomy for testing the Simultaneous Level Planarity of a set of level graphs, with respect to both the number of level graphs and the number of levels.

Level Planarity: Transitivity vs. Even Crossings

“…The novel connection has further led us to a new testing algorithm for radial level planarity. Can similar results be achieved for level planarity on a rolling cylinder or on a torus [1]?…”

Simultaneous FPQ-Ordering and Hybrid Planarity Testing

“…An instance I = (N, Z) of Simultaneous PQ-Ordering is normalized if, for each arc (T i , T j ; ϕ) ∈ Z and for each internal node µ j ∈ T j , tree T i contains exactly one node µ i that is fixed by µ j . Every instance of Simultaneous PQ-Ordering can be normalized by means of an operation called the normalization [2,7], which is defined as follows. Consider each arc (T i , T j ; ϕ) ∈ Z and replace T j with T i | ϕ(L(Tj )) ∩ T j in I, that is, replace tree T j with its intersection with the projection of its parent T i to the set of leaves of T i obtained by applying mapping ϕ to the leaves L(T j ) of T j .…”

Level-Planar Drawings with Few Slopes

“…We refer the reader to [5] for an older literature overview. The problem has been considered for orthogonal drawings [2] and level-planar drawings [3]. Up to our knowledge, neither partial nor simultaneous drawings have been considered in the restricted slope setting.…”