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. An earlier version of this paper appeared in the proceedings of WG 2011. The first author gratefully acknowledges support from the Swiss National Science Foundation Grant No. 200021-125287/1. The second author gratefully acknowledges the support from NSA Grant H98230-08-10043 and the Swiss National Science Foundation Grant No. 200021-125287/1.