Towards the Hanani-Tutte Theorem for Clustered Graphs

  • Radoslav FulekEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)


The weak variant of the Hanani–Tutte theorem says that a graph is planar, if it can be drawn in the plane so that every pair of edges cross an even number of times. Moreover, we can turn such a drawing into an embedding without changing the order in which edges leave the vertices. We prove a generalization of the weak Hanani–Tutte theorem that also easily implies the monotone variant of the weak Hanani–Tutte theorem by Pach and Tóth. Thus, our result can be thought of as a common generalization of these two neat results. In other words, we prove the weak Hanani-Tutte theorem for strip clustered graphs, whose clusters are linearly ordered vertical strips in the plane and edges join only vertices in the same cluster or in neighboring clusters with respect to this order.

Besides usual tools for proving Hanani-Tutte type results our proof combines Hall’s marriage theorem, and a characterization of embedded upward planar digraphs due to Bertolazzi et al.


Hanani–Tutte theorem Hall’s theorem Upward planarity C-planarity 



We would like to express our special thanks of gratitude to the organizers and participants of the 11th GWOP workshop, where we could discuss the research problems treated in the present paper. In particular, we especially benefited from the discussions with Bettina Speckmann, Edgardo Roldán-Pensado and Sebastian Stich. Furthermore, we would like to thank Ján Kynčl for useful discussions at the initial stage, and Gábor Tardos for comments at the final stage of this work.


  1. 1.
    Angelini, P., Da Lozzo, G., Di Battista, G., Frati, F.: Strip planarity testing. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 37–48. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  2. 2.
    Bertolazzi, P., Di Battista, G., Liotta, G., Mannino, C.: Upward drawings of triconnected digraphs. Algorithmica 12(6), 476–497 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cairns, G., Nikolayevsky, Y.: Bounds for generalized thrackles. Discrete Comput. Geom. 23(2), 191–206 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cortese, P.F., Di Battista, G., Patrignani, M., Pizzonia, M.: Clustering cycles into cycles of clusters. J. Graph Algorithms Appl. 9(3), 391–413 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cortese, P.F., Di Battista, G.: Clustered planarity (invited lecture). In: Twenty-first Annual Symposium on Computational Geometry (proc. SoCG 05), pp. 30–32. ACM (2005)Google Scholar
  6. 6.
    Diestel, R.: Graph Theory. Springer, New York (2010)CrossRefGoogle Scholar
  7. 7.
    Feng, Q.-W., Cohen, R.F., Eades, R.: How to draw a planar clustered graph. In: Li, M., Du, D.-Z. (eds.) COCOON 1995. LNCS, vol. 959, pp. 21–30. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  8. 8.
    Feng, Q.-W., Cohen, R.F., Eades, P.: Planarity for clustered graphs. In: Spirakis, P.G. (ed.) ESA 1995. LNCS, vol. 979, pp. 213–226. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  9. 9.
    Fulek, R., Kynčl, J., Malinović, I., Pálvölgyi, D.: Efficient c-planarity testing algebraically. arXiv:1305.4519
  10. 10.
    Fulek, R., Pelsmajer, M., Schaefer, M., Štefankovič, D.: Hanani-Tutte, monotone drawings and level-planarity. In: Pach, J. (ed.) Thirty Essays in Geometric Graph Theory, pp. 263–288. Springer, New York (2012)Google Scholar
  11. 11.
    Gortler, S.J., Gotsman, C., Thurston, D.: Discrete one-forms on meshes and applications to 3D mesh parameterization. J. CAGD 23, 83–112 (2006)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Guillemin, V., Pollack, A.: Differential Topology. Prentice-Hall (1974)Google Scholar
  13. 13.
    Hanani, H.: Über wesentlich unplättbare Kurven im drei-dimensionalen Raume. Fundam. Math. 23, 135–142 (1934)Google Scholar
  14. 14.
    Pach, J., Tóth, G.: Which crossing number is it anyway? J. Combin. Theory Ser. B 80(2), 225–246 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pach, J., Tóth, J.: Monotone drawings of planar graphs. J. Graph Theory 46(1), 39–47 (2004). version)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pelsmajer, M.J., Schaefer, M., Stasi, D.: Strong Hanani-Tutte on the projective plane. SIAM J. Discrete Math. 23(3), 1317–1323 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings. J. Combin. Theory Ser. B 97(4), 489–500 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings on surfaces. Eur. J. Comb. 30(7), 1704–1717 (2009)CrossRefzbMATHGoogle Scholar
  19. 19.
    Schaefer, M.: Hanani-Tutte and related results. Bolyai Memorial Volume (2011)Google Scholar
  20. 20.
    Schaefer, M.: Toward a theory of planarity: Hanani-Tutte and planarity variants. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 162–173. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  21. 21.
    Schaefer, M., Štefankovič, D.: Block additivity of \(\mathbb{Z}\) \(_\text{2 }\)-embeddings. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 185–195. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  22. 22.
    Tutte, W.T.: Toward a theory of crossing numbers. J. Combin. Theory 8, 45–53 (1970)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Industrial Engineering and Operations ResearchColumbia UniversityNew York CityUSA

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