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.
CITATION STYLE
Fulek, R., Pelsmajer, M. J., Schaefer, M., & Štefankovič, D. (2013). Hanani-Tutte, monotone drawings, and level-planarity. In Thirty Essays on Geometric Graph Theory (Vol. 9781461401100, pp. 263–287). Springer New York. https://doi.org/10.1007/978-1-4614-0110-0_14
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