International Symposium on Graph Drawing and Network Visualization

Graph Drawing and Network Visualization pp 99-110 | Cite as

Hanani-Tutte for Radial Planarity

  • Radoslav Fulek
  • Michael Pelsmajer
  • Marcus Schaefer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)

Abstract

A drawing of a graph G is radial if the vertices of G are placed on concentric circles \(C_1, \ldots , C_k\) with common center c, and edges are drawn radially: every edge intersects every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling.

We show that a graph G is radial planar if G has a radial drawing in which every two edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the weak variant of the Hanani-Tutte theorem for radial planarity. This generalizes a result by Pach and Tóth.

References

  1. 1.
    Bachmaier, C., Brandenburg, F.J., Forster, M.: Radial level planarity testing and embedding in linear time. J. Graph Algorithms Appl. 9, 2005 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cairns, G., Nikolayevsky, Y.: Bounds for generalized thrackles. Discrete Comput. Geom. 23(2), 191–206 (2000)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chimani, M., Zeranski, R.: Upward planarity testing: a computational study. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 13–24. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  5. 5.
    Chojnacki, C., Hanani, H.: Über wesentlich unplättbare Kurven im dreidimensionalen Raume. Fundamenta Mathematicae 23, 135–142 (1934)Google Scholar
  6. 6.
    Di Battista, G., Nardelli, E.: Hierarchies and planarity theory. IEEE Trans. Syst. Man Cybern. 18(6), 1035–1046 (1989)CrossRefGoogle Scholar
  7. 7.
    Di Giacomo, E., Didimo, W., Liotta, G.: Spine and radial drawings, chapter 8. In: Roberto, T. (ed.) Handbook of Graph Drawing and Visualization. Discrete Mathematics and Its Applications. Chapman and Hall/CRC, Boca Raton (2013)Google Scholar
  8. 8.
    Fulek, R., Kynčl, J., Malinović, I., Pálvölgyi, D.: Clustered planarity testing revisited. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 428–439. Springer, Heidelberg (2014) Google Scholar
  9. 9.
    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, New York (2013)CrossRefGoogle Scholar
  10. 10.
    Gross, J.L., Tucker, T.W.: Topological Graph Theory. Dover Publications Inc., Mineola (2001). Reprint of the 1987 originalMATHGoogle Scholar
  11. 11.
    Gutwenger, C., Mutzel, P., Schaefer, M.: Practical experience with Hanani-Tutte for testing \(c\)-planarity. In: McGeoch, C.C., Meyer, U. (eds.) 2014 Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments (ALENEX), pp. 86–97. SIAM, Portland (2014)CrossRefGoogle Scholar
  12. 12.
    Jünger, M., Leipert, S.: Level planar embedding in linear time. J. Graph Algorithms Appl. 6(1), 72–81 (2002)CrossRefGoogle Scholar
  13. 13.
    Northway, M.L.: A method for depicting social relationships obtained by sociometric testing. Sociometry 3(2), 144–150 (1940)CrossRefGoogle Scholar
  14. 14.
    Pach, J., Tóth, G.: Monotone drawings of planar graphs. J. Graph Theory 46(1), 39–47 (2004). Updated version: arXiv:1101.0967 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Pelsmajer, M.J., Schaefer, M., Stasi, D.: Strong Hanani-Tutte on the projective plane. SIAM J. Discrete Math. 23(3), 1317–1323 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings. J. Combin. Theor. Ser. B 97(4), 489–500 (2007)CrossRefMATHGoogle Scholar
  17. 17.
    Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing even crossings on surfaces. Eur. J. Comb. 30(7), 1704–1717 (2009)CrossRefMATHGoogle Scholar
  18. 18.
    Schaefer, M.: Toward a theory of planarity: Hanani-Tutte and planarity variants. J. Graph Algortihms Appl. 17(4), 367–440 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Schaefer, M.: Hanani-Tutte and related results. In: Bárány, I., Böröczky, K.J., Tóth, G.F., Pach, J. (eds.) A Tribute to László Fejes Tóth. Bolyai Society Mathematical Studies, vol. 24, pp. 259–299. Springer, Berlin (2014)Google Scholar
  20. 20.
    Tutte, W.T.: Toward a theory of crossing numbers. J. Comb. Theor. 8, 45–53 (1970)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Wiedemann, D.H.: Solving sparse linear equations over finite fields. IEEE Trans. Inf. Theor. 32(1), 54–62 (1986)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Radoslav Fulek
    • 1
  • Michael Pelsmajer
    • 2
  • Marcus Schaefer
    • 3
  1. 1.IST AustriaKlosterneuburgAustria
  2. 2.Illinois Institute of TechnologyChicagoUSA
  3. 3.DePaul UniversityChicagoUSA

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