[Paper Review] Beyond Level Planarity

Patrizio Angelini, Giordano Da Lozzo+4 more|arXiv (Cornell University)|Oct 28, 2015

Advanced Graph Theory Research21 references3 citations

TL;DR

This paper establishes that Cyclic Level Planarity and Torus Level Planarity are polynomial-time solvable, resolving two long-standing open problems in graph drawing. The authors introduce a reduction-based approach that transforms simultaneous level planarity instances into cyclic level planarity problems, enabling efficient algorithms for specific cases, including two-level graphs with opposing edge directions.

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 $ extit{Cyclic Level Planarity}$ and $ extit{Torus Level Planarity}$, are polynomial-time solvable. Moreover, we show a complexity dichotomy for testing the $ extit{Simultaneous Level Planarity}$ of a set of level graphs, with respect to both the number of level graphs and the number of levels.

Motivation & Objective

• To resolve the computational complexity of Cyclic Level Planarity and Torus Level Planarity, two open problems in graph drawing.
• To establish a complexity dichotomy for Simultaneous Level Planarity with respect to the number of level graphs and levels.
• To extend the notion of level planarity to surfaces beyond the plane, particularly the rolling cylinder and torus.
• To provide a polynomial-time algorithm for testing level embeddings on the rolling cylinder and torus surfaces.
• To introduce and analyze a new variant of simultaneous level planarity for multiple level graphs on the same surface.

Proposed method

• Reducing Simultaneous Level Planarity instances with two graphs on two levels to a single Cyclic Level Planarity instance by reversing edge directions in one graph.
• Constructing a proper level graph where edges from one graph go from level 1 to 2 and edges from the other go from 2 to 1, forming a cyclic structure.
• Using a reduction to Cyclic Level Planarity to leverage existing polynomial-time algorithms for the rolling cylinder surface.
• Proving equivalence between simultaneous level planarity and cyclic level planarity via topological embedding transformations on the rolling cylinder surface.
• Applying known linear-time algorithms for Level Planarity to the transformed instance, ensuring polynomial-time solvability.
• Extending the framework to show that connectedness can be preserved with one additional level, and that geometric and fixed-edge simultaneous embeddings are also tractable.

Experimental results

Research questions

• RQ1 Is Cyclic Level Planarity, the problem of testing level planarity on the rolling cylinder, computationally tractable?
• RQ2 Can Torus Level Planarity, the level planarity variant on the torus surface, be decided in polynomial time?
• RQ3 What is the computational complexity of Simultaneous Level Planarity when parameterized by the number of level graphs and the number of levels?
• RQ4 Can simultaneous level planarity for two-level graphs be solved efficiently, particularly when edges flow in opposite directions across levels?
• RQ5 Is there a meaningful generalization of level planarity to surfaces of genus greater than one, and how can such a framework be formalized?

Key findings

• Cyclic Level Planarity is polynomial-time solvable, resolving a long-standing open problem in graph drawing.
• Torus Level Planarity is also polynomial-time solvable, establishing that level planarity extends efficiently to the torus surface.
• A reduction from Simultaneous Level Planarity for two graphs on two levels to Cyclic Level Planarity enables a quadratic-time algorithm.
• The Simultaneous Level Planarity problem exhibits a complexity dichotomy: it is polynomial-time solvable when the number of levels or graphs is bounded, but NP-hard otherwise.
• The proposed reduction preserves connectivity when one additional level is added, and extends to geometric and fixed-edge simultaneous embeddings.
• The results imply that level planarity on the rolling cylinder and torus can be tested in quartic time, though tighter bounds remain an open challenge.