Hanani-Tutte and Monotone Drawings
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.
KeywordsPlanar Graph Rotation System Swiss National Science Foundation Interior Vertex Independent Edge
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- 3.Di Battista, G., Nardelli, E.: Hierarchies and planarity theory. IEEE Trans. Systems Man Cybernet. 18(6), 1035–1046 (1988, 1989)Google Scholar
- 6.Chojnacki, C., (Hanani, H.).: Über wesentlich unplättbare Kurven im drei-dimensionalen Raume. Fundamenta Mathematicae 23, 135–142 (1934)Google Scholar
- 9.Matoušek, J.: Using the Borsuk-Ulam theorem. Universitext. Springer, Berlin (2003); Lectures on topological methods in combinatorics and geometry, Written in cooperation with Anders Björner and Günter M. ZieglerGoogle Scholar
- 10.Matousek, J., Tancer, M., Wagner, U.: Hardness of embedding simplicial complexes in ℝd. In: Mathieu, C. (ed.) Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, New York, NY, USA, January 4-6, pp. 855–864. SIAM (2009)Google Scholar
- 13.Pach, J., Tóth Monotone, G.: Drawings of planar graphs. ArXiv e-prints (January 2011)Google Scholar
- 14.Pach, J., Tóth, G.: Monotone crossing number. In: Graph Drawing (to appear, 2011)Google Scholar
- 18.Schaefer, M.: Hanani-Tutte and related results. To appear in Bolyai Memorial VolumeGoogle Scholar