Toward a Theory of Planarity: Hanani-Tutte and Planarity Variants

  • Marcus Schaefer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)

Abstract

We study Hanani-Tutte style theorems for various notions of planarity, including partially embedded planarity, and simultaneous planarity. This approach brings together the combinatorial, computational and algebraic aspects of planarity notions and may serve as a uniform foundation for planarity, as suggested in the writings of Tutte and Wu.

References

  1. 1.
    Angelini, P., Di Battista, G., Frati, F., Patrignani, M., Rutter, I.: Testing the Simultaneous Embeddability of Two Graphs Whose Intersection Is a Biconnected Graph or a Tree. In: Iliopoulos, C.S., Smyth, W.F. (eds.) IWOCA 2010. LNCS, vol. 6460, pp. 212–225. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Angelini, P., Di Battista, G., Frati, F.: Simultaneous Embedding of Embedded Planar Graphs. In: Asano, T., Nakano, S.-I., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 271–280. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Angelini, P., Di Battista, G., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. In: Charikar, M. (ed.) SODA 2010, pp. 202–221. SIAM (2010)Google Scholar
  4. 4.
    Angelini, P., Di Battista, G., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. In: Charikar, M. (ed.) Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, pp. 202–221. SIAM (2010)Google Scholar
  5. 5.
    Archdeacon, D.: A Kuratowski theorem for the projective plane. J. Graph Theory 5(3), 243–246 (1981)MathSciNetMATHCrossRefGoogle 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 (electronic) (2005)Google Scholar
  7. 7.
    Bläsius, T., Kobourov, S.G., Rutter, I.: Simutlaneous Embeedings of Planar Graphs. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization. CRC Press (to appear)Google Scholar
  8. 8.
    Bläsius, T., Rutter, I.: Simultaneous PQ-ordering with applications to constrained embedding problems. CoRR abs/1112.0245 (2011)Google Scholar
  9. 9.
    Bläsius, T., Rutter, I.: Disconnectivity and Relative Positions in Simultaneous Embeddings. ArXiv e-prints (April 2012)Google Scholar
  10. 10.
    Chartrand, G., Harary, F.: Planar permutation graphs. Ann. Inst. H. Poincaré Sect. B (N.S.) 3, 433–438 (1967)MathSciNetMATHGoogle Scholar
  11. 11.
    Chimani, M., Jünger, M., Schulz, M.: Crossing minimization meets simultaneous drawing. In: Visualization Symposium, PacificVIS 2008, pp. 33–40. IEEE (2008)Google Scholar
  12. 12.
    Chojnacki (Haim Hanani), C.: Über wesentlich unplättbare Kurven im drei-dimensionalen Raume. Fundamenta Mathematicae 23, 135–142 (1934)Google Scholar
  13. 13.
    Chung, F.R.K., Leighton, F.T., Rosenberg, A.L.: Embedding graphs in books: a layout problem with applications to VLSI design. In: Graph Theory with Applications to Algorithms and Computer Science (Kalamazoo, Mich., 1984), pp. 175–188. Wiley-Intersci. Publ., Wiley, New York (1985)Google Scholar
  14. 14.
    Cortese, P.F., Di Battista, G.: Clustered planarity. In: Computational Geometry, SCG 2005, pp. 32–34. ACM, New York (2005)Google Scholar
  15. 15.
    Feng, Q.W.: Algorithms for Drawing Clustered Graphs. Ph.D. thesis, Department of Computer Science and Software engineering, University of Newclastle (April 1997)Google Scholar
  16. 16.
    Feng, Q.W., Cohen, R.F., Eades, P.: 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
  17. 17.
    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
  18. 18.
    Fulek, R., Pelsmajer, M., 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 (2012) Google Scholar
  19. 19.
    Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31(2), 601–625 (electronic) (2001)Google Scholar
  20. 20.
    Gassner, E., Jünger, M., Percan, M., Schaefer, M., Schulz, M.: Simultaneous Graph Embeddings with Fixed Edges. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 325–335. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  21. 21.
    Glover, H.H., Huneke, J.P., Wang, C.S.: 103 graphs that are irreducible for the projective plane. J. Combin. Theory Ser. B 27(3), 332–370 (1979)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Haeupler, B., Jampani, K.R., Lubiw, A.: Testing Simultaneous Planarity When the Common Graph Is 2-Connected. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part II. LNCS, vol. 6507, pp. 410–421. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  23. 23.
    Haeupler, B., Jampani, K., Lubiw, A.: Testing simultaneous planarity when the common graph is 2-connected. CoRR abs/1009.4517 (2010)Google Scholar
  24. 24.
    Hopcroft, J., Tarjan, R.: Efficient planarity testing. J. Assoc. Comput. Mach. 21, 549–568 (1974)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Jelínek, V., Kratochvíl, J., Rutter, I.: A Kuratowski-type theorem for planarity of partially embedded graphs. In: Hurtado, F., van Kreveld, M.J. (eds.) SoCG 2011, pp. 107–116. ACM (2011)Google Scholar
  26. 26.
    Jünger, M., Leipert, S., Mutzel, P.: Level Planarity Testing in Linear Time. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 224–237. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  27. 27.
    Kuratowski, C.: Sur les problèmes des courbes gauches en Topologie. Fund. Math. 15, 271–283 (1930)MATHGoogle Scholar
  28. 28.
    Pach, J., Tóth, G.: Monotone drawings of planar graphs. J. Graph Theory 46(1), 39–47 (2004)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Pach, J., Tóth, G.: Monotone drawings of planar graphs. ArXiv e-prints (January 2011)Google Scholar
  30. 30.
    Patrignani, M.: On extending a partial straight-line drawing. Internat. J. Found. Comput. Sci. 17(5), 1061–1069 (2006)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Pelsmajer, M.J., Schaefer, M., Stasi, D.: Strong Hanani–Tutte on the projective plane. SIAM Journal on Discrete Mathematics 23(3), 1317–1323 (2009)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing independently even crossings. SIAM Journal on Discrete Mathematics 24(2), 379–393 (2010)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Schaefer, M.: Hanani-Tutte and related results, to appear in Bolyai Memorial VolumeGoogle Scholar
  34. 34.
    Tamassia, R.: Handbook of Graph Drawing and Visualization. Discrete Mathematics and Its Applications. Chapman and Hall (2012) (to appear)Google Scholar
  35. 35.
    Tutte, W.T.: Toward a theory of crossing numbers. J. Combinatorial Theory 8, 45–53 (1970)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Marcus Schaefer
    • 1
  1. 1.School of ComputingDePaul UniversityChicagoUSA

Personalised recommendations