Hanani–Tutte, Monotone Drawings, and Level-Planarity
A drawing of a graph is x-monotone if every edge intersects every vertical line at most once and every vertical line contains at most one vertex. Pach and Tóth showed that if a graph has an x-monotone drawing in which every pair of edges crosses an even number of times, then the graph has an x-monotone embedding in which the x-coordinates of all vertices are unchanged. We give a new proof of this result and strengthen it by showing that the conclusion remains true even if adjacent edges are allowed to cross each other oddly. This answers a question posed by Pach and Tóth. We show that a further strengthening to a “removing even crossings” lemma is impossible by separating monotone versions of the crossing and the odd crossing number.
Our results extend to level-planarity, which is a well-studied generalization of x-monotonicity. We obtain a new and simple algorithm to test level-planarity in quadratic time, and we show that x-monotonicity of edges in the definition of level-planarity can be relaxed.
- Hanani–Tutte, Monotone Drawings, and Level-Planarity
- Book Title
- Thirty Essays on Geometric Graph Theory
- pp 263-287
- Print ISBN
- Online ISBN
- Springer New York
- Copyright Holder
- Springer Science+Business Media New York
- Additional Links
- eBook Packages
- János Pach (11) (12)
- Editor Affiliations
- 11. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences
- 12. École Polytechnique Fédérale de Lausanne
- Author Affiliations
- 1. Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
- 2. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, 60616, USA
- 3. Department of Computer Science, DePaul University, Chicago, IL, 60604, USA
- 4. Computer Science Department, University of Rochester, Rochester, NY, 14627-0226, USA
To view the rest of this content please follow the download PDF link above.