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A Direct Proof of the Strong Hanani–Tutte Theorem on the Projective Plane

  • Éric Colin de Verdière
  • Vojtěch KalužaEmail author
  • Pavel Paták
  • Zuzana Patáková
  • Martin Tancer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)

Abstract

We reprove the strong Hanani–Tutte theorem on the projective plane. In contrast to the previous proof by Pelsmajer, Schaefer and Stasi, our method is constructive and does not rely on the characterization of forbidden minors, which gives hope to extend it to other surfaces. Moreover, our approach can be used to provide an efficient algorithm turning a Hanani–Tutte drawing on the projective plane into an embedding.

Keywords

Graph drawing Graph embedding Hanani–Tutte theorem Projective plane Topological graph theory 

Notes

Acknowledgment

We would like to thank Alfredo Hubard for fruitful discussions and valuable comments.

References

  1. 1.
    Cairns, G., Nikolayevsky, Y.: Bounds for generalized thrackles. Discret. Comput. Geom. 23(2), 191–206 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chojnacki, C.: Über wesentlich unplättbare Kurven im dreidimensionalen Raume. Fundamenta Mathematicae 23(1), 135–142 (1934)MathSciNetzbMATHGoogle Scholar
  3. 3.
    de Fraysseix, H., Ossona de Mendez, P.: Trémaux trees and planarity. Eur. J. Comb. 33(3), 279–293 (2012)CrossRefzbMATHGoogle Scholar
  4. 4.
    de Fraysseix, H., Rosenstiehl, P.: A characterization of planar graphs by Trémaux orders. Combinatorica 5(2), 127–135 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2010)CrossRefzbMATHGoogle Scholar
  6. 6.
    Fulek, R., Kynčl, J., Malinović, I., Pálvölgyi, D.: Clustered planarity testing revisited. Electron. J. Comb. 22(4), P4–P24 (2015)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Fulek, R., Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Adjacent crossings do matter. J. Graph Algorithms Appl. 16(3), 759–782 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hopcroft, J., Tarjan, R.: Efficient planarity testing. J. ACM 21(4), 549–568 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kawarabayashi, K., Mohar, B., Reed, B.: A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded tree-width. In: 49th Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 771–780, October 2008Google Scholar
  10. 10.
    Kuratowski, C.: Sur le problème des courbes gauches en Topologie. Fundamenta Mathematicae 15(1), 271–283 (1930)zbMATHGoogle Scholar
  11. 11.
    Levow, R.B.: On Tutte’s algebraic approach to the theory of crossing numbers. In: Proceedings of the Third Southeastern Conference on Combinatorics, Graph Theory, and Computing, pp. 314–315. Florida Atlantic Univ., Boca Raton (1972)Google Scholar
  12. 12.
    Mohar, B.: A linear time algorithm for embedding graphs in an arbitrary surface. SIAM J. Discret. Math. 12(1), 6–26 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (2001)zbMATHGoogle Scholar
  14. 14.
    Pelsmajer, M.J., Schaefer, M., Stasi, D.: Strong Hanani-Tutte on the projective plane. SIAM J. Discret. Math. 23(3), 1317–1323 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings. J. Comb. Theory Ser. B 97(4), 489–500 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings on surfaces. Electron. Notes Discret. Math. 29, 85–90 (2007). European Conference on Combinatorics, Graph Theory and ApplicationsCrossRefzbMATHGoogle Scholar
  17. 17.
    Schaefer, M.: Hanani-Tutte and related results. In: Bárány, I., Böröczky, K.J., Tóth, G.F., Pach, J. (eds.) Geometry – Intuitive. Discrete, and Convex: A Tribute to László Fejes Tóth, pp. 259–299. Springer, Heidelberg (2013)Google Scholar
  18. 18.
    Schaefer, M.: Toward a theory of planarity: Hanani-Tutte and planarity variants. J. Graph Algorithms Appl. 17(4), 367–440 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tutte, W.T.: Toward a theory of crossing numbers. J. Comb. Theory 8(1), 45–53 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    van Kampen, E.R.: Komplexe in euklidischen Räumen. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 9(1), 72–78 (1933)CrossRefzbMATHGoogle Scholar
  21. 21.
    Wu, W.: On the planar imbedding of linear graphs I. J. Syst. Sci. Math. Sci. 5(4), 290–302 (1985)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Éric Colin de Verdière
    • 1
  • Vojtěch Kaluža
    • 2
    Email author
  • Pavel Paták
    • 3
  • Zuzana Patáková
    • 3
  • Martin Tancer
    • 2
  1. 1.CNRS and Département d’informatique, École normale supérieureParisFrance
  2. 2.Department of Applied MathematicsCharles UniversityPragueCzech Republic
  3. 3.Einstein Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

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