Hanani-Tutte and Monotone Drawings

  • Radoslav Fulek
  • Michael J. Pelsmajer
  • Marcus Schaefer
  • Daniel Štefankovič
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6986)


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.


Planar Graph Rotation System Swiss National Science Foundation Interior Vertex Independent Edge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Bienstock, D., Dean, N.: Bounds for rectilinear crossing numbers. J. Graph Theory 17(3), 333–348 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cairns, G., Nikolayevsky, Y.: Bounds for generalized thrackles. Discrete Comput. Geom. 23(2), 191–206 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Di Battista, G., Nardelli, E.: Hierarchies and planarity theory. IEEE Trans. Systems Man Cybernet. 18(6), 1035–1046 (1988, 1989)Google Scholar
  4. 4.
    Eades, P., Feng, Q., Lin, X., Nagamochi, H.: Straight-line drawing algorithms for hierarchical graphs and clustered graphs. Algorithmica 44(1), 1–32 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    Chojnacki, C., (Hanani, H.).: Über wesentlich unplättbare Kurven im drei-dimensionalen Raume. Fundamenta Mathematicae 23, 135–142 (1934)Google Scholar
  7. 7.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Lin, X., Eades, P.: Towards area requirements for drawing hierarchically planar graphs. Theor. Comput. Sci. 292(3), 679–695 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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. 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
  11. 11.
    Pach, J., Tóth, G.: Which crossing number is it anyway? J. Combin. Theory Ser. B 80(2), 225–246 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pach, J., Tóth, G.: Monotone drawings of planar graphs. J. Graph Theory 46(1), 39–47 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pach, J., Tóth Monotone, G.: Drawings of planar graphs. ArXiv e-prints (January 2011)Google Scholar
  14. 14.
    Pach, J., Tóth, G.: Monotone crossing number. In: Graph Drawing (to appear, 2011)Google Scholar
  15. 15.
    Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings. J. Combin. Theory Ser. B 97(4), 489–500 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing independently even crossings. SIAM Journal on Discrete Mathematics 24(2), 379–393 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Schaefer, M.: Hanani-Tutte and related results. To appear in Bolyai Memorial VolumeGoogle Scholar
  19. 19.
    Tutte, W.T.: Toward a theory of crossing numbers. J. Combinatorial Theory 8, 45–53 (1970)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Radoslav Fulek
    • 1
  • Michael J. Pelsmajer
    • 2
  • Marcus Schaefer
    • 3
  • Daniel Štefankovič
    • 4
  1. 1.Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Illinois Institute of TechnologyChicagoUSA
  3. 3.DePaul UniversityChicagoUSA
  4. 4.University of RochesterRochesterUSA

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