Hanani-Tutte and monotone drawings

Authors:
Radoslav Fulek;Michael J. Pelsmajer;Marcus Schaefer;Daniel Štefankovič
Affiliations:
Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland;Illinois Institute of Technology, Chicago, IL;DePaul University, Chicago, IL;University of Rochester, Rochester, NY
Venue:
WG'11 Proceedings of the 37th international conference on Graph-Theoretic Concepts in Computer Science
Year:
2011

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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.