Fulek, RadoslavPelsmajer, Michael J.Schaefer, MarcusStefankovic, Daniel2013-02-282013-02-282013-02-28201110.1007/978-3-642-25870-1_26https://infoscience.epfl.ch/handle/20.500.14299/90009WOS:000307088100026A 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 Toth 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 oddly. This answers a question posed by Pach and Toth. Moreover, we show that an extension of this result for graphs with non-adjacent pairs of edges crossing oddly fails even if there exists only one such pair in a graph.text::conference output::conference proceedings::conference paper