Abstract
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 oddly. This answers a question posed by Pach and Tóth. 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.
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Fulek, R., Pelsmajer, M.J., Schaefer, M., Štefankovič, D. (2011). Hanani-Tutte and Monotone Drawings. In: Kolman, P., Kratochvíl, J. (eds) Graph-Theoretic Concepts in Computer Science. WG 2011. Lecture Notes in Computer Science, vol 6986. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25870-1_26
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DOI: https://doi.org/10.1007/978-3-642-25870-1_26
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