Level Planarity: Transitivity vs. Even Crossings

  • Guido Brückner
  • Ignaz Rutter
  • Peter StumpfEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11282)


Recently, Fulek et al. [1, 2, 3] have presented Hanani-Tutte results for (radial) level planarity, i.e., a graph is (radial) level planar if it admits a (radial) level drawing where any two (independent) edges cross an even number of times. We show that the 2-Sat formulation of level planarity testing due to Randerath et al. [4] is equivalent to the strong Hanani-Tutte theorem for level planarity [3]. Further, we show that this relationship carries over to radial level planarity, which yields a novel polynomial-time algorithm for testing radial level planarity.


  1. 1.
    Fulek, R., Pelsmajer, M., Schaefer, M.: Hanani-Tutte for radial planarity. In: Di Giacomo, E., Lubiw, A. (eds.) GD 2015. LNCS, vol. 9411, pp. 99–110. Springer, Cham (2015). Scholar
  2. 2.
    Fulek, R., Pelsmajer, M., Schaefer, M.: Hanani-Tutte for radial planarity II. In: Hu, Y., Nöllenburg, M. (eds.) GD 2016. LNCS, vol. 9801, pp. 468–481. Springer, Cham (2016). Scholar
  3. 3.
    Fulek, R., Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Hanani-Tutte, monotone drawings, and level-planarity. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 263–287. Springer, New York (2013). Scholar
  4. 4.
    Randerath, B., et al.: A satisfiability formulation of problems on level graphs. Electron. Notes Discrete Math. 9, 269–277 (2001). lICS 2001 Workshop on Theory and Applications of Satisfiability Testing (SAT 2001)CrossRefGoogle Scholar
  5. 5.
    Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31(2), 601–625 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bachmaier, C., Brandenburg, F.J., Forster, M.: Radial level planarity testing and embedding in linear time. J. Graph Algorithms Appl. 9(1), 53–97 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Jünger, M., Leipert, S.: Level planar embedding in linear time. In: Kratochvíyl, J. (ed.) GD 1999. LNCS, vol. 1731, pp. 72–81. Springer, Heidelberg (1999). Scholar
  8. 8.
    Harrigan, M., Healy, P.: Practical level planarity testing and layout with embedding constraints. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 62–68. Springer, Heidelberg (2008). Scholar
  9. 9.
    Brückner, G., Rutter, I.: Partial and constrained level planarity. In: Klein, P.N. (ed.) Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 2000–2011. SIAM (2017)Google Scholar
  10. 10.
    Klemz, B., Rote, G.: Ordered level planarity, geodesic planarity and bi-monotonicity. In: Frati, F., Ma, K.-L. (eds.) GD 2017. LNCS, vol. 10692, pp. 440–453. Springer, Cham (2018). Scholar
  11. 11.
    Chojnacki, C.: Über wesentlich unplättbare Kurven im dreidimensionalen Raume. Fundam. Math. 23(1), 135–142 (1934)CrossRefGoogle Scholar
  12. 12.
    Tutte, W.T.: Toward a theory of crossing numbers. J. Combin. Theory 8(1), 45–53 (1970)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Pach, J., Tóth, G.: Monotone drawings of planar graphs. J. Graph Theory 46(1), 39–47 (2004)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Pach, J., Tóth, G.: monotone drawings of planar graphs. ArXiv:1101.0967 e-prints (2011)
  15. 15.
    Brückner, G., Rutter, I., Stumpf, P.: Level planarity: transitivity vs. even crossings. CoRR abs/1808.09931 (2018)Google Scholar
  16. 16.
    Angelini, P., Da Lozzo, G., Di Battista, G., Frati, F., Patrignani, M., Rutter, I.: Beyond level planarity. In: Hu, Y., Nöllenburg, M. (eds.) GD 2016. LNCS, vol. 9801, pp. 482–495. Springer, Cham (2016). Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Faculty of InformaticsKarlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.Department of Computer Science and MathematicsUniversity of PassauPassauGermany

Personalised recommendations