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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bienstock, D., Dean, N.: Bounds for rectilinear crossing numbers. J. Graph Theory 17(3), 333–348 (1993)
Cairns, G., Nikolayevsky, Y.: Bounds for generalized thrackles. Discrete Comput. Geom. 23(2), 191–206 (2000)
Di Battista, G., Nardelli, E.: Hierarchies and planarity theory. IEEE Trans. Systems Man Cybernet. 18(6), 1035–1046 (1988, 1989)
Eades, P., Feng, Q., Lin, X., Nagamochi, H.: Straight-line drawing algorithms for hierarchical graphs and clustered graphs. Algorithmica 44(1), 1–32 (2006)
Estrella-Balderrama, A., Fowler, J.J., Kobourov, S.G.: On the Characterization of Level Planar Trees by Minimal Patterns. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 69–80. Springer, Heidelberg (2010)
Chojnacki, C., (Hanani, H.).: Über wesentlich unplättbare Kurven im drei-dimensionalen Raume. Fundamenta Mathematicae 23, 135–142 (1934)
Kleitman, D.J.: A note on the parity of the number of crossings of a graph. J. Combinatorial Theory Ser. B 21(1), 88–89 (1976)
Lin, X., Eades, P.: Towards area requirements for drawing hierarchically planar graphs. Theor. Comput. Sci. 292(3), 679–695 (2003)
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. Ziegler
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)
Pach, J., Tóth, G.: Which crossing number is it anyway? J. Combin. Theory Ser. B 80(2), 225–246 (2000)
Pach, J., Tóth, G.: Monotone drawings of planar graphs. J. Graph Theory 46(1), 39–47 (2004)
Pach, J., Tóth Monotone, G.: Drawings of planar graphs. ArXiv e-prints (January 2011)
Pach, J., Tóth, G.: Monotone crossing number. In: Graph Drawing (to appear, 2011)
Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings. J. Combin. Theory Ser. B 97(4), 489–500 (2007)
Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Odd crossing number and crossing number are not the same. Discrete Comput. Geom. 39(1), 442–454 (2008)
Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing independently even crossings. SIAM Journal on Discrete Mathematics 24(2), 379–393 (2010)
Schaefer, M.: Hanani-Tutte and related results. To appear in Bolyai Memorial Volume
Tutte, W.T.: Toward a theory of crossing numbers. J. Combinatorial Theory 8, 45–53 (1970)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-642-25870-1_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25869-5
Online ISBN: 978-3-642-25870-1
eBook Packages: Computer ScienceComputer Science (R0)