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.
We thank the anonymous referee for careful proofreading.
Radoslav Fulek gratefully acknowledges support from the Swiss National Science Foundation Grant No. 200021-125287/1.
Michael J. Pelsmajer gratefully acknowledges the support from NSA Grant H98230-08-1-0043 and the Swiss National Science Foundation Grant No. 200021-125287/1.
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