Counting Plane Graphs: Flippability and Its Applications

  • Michael Hoffmann
  • André Schulz
  • Micha Sharir
  • Adam Sheffer
  • Csaba D. Tóth
  • Emo Welzl
Chapter

Abstract

We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to pseudo-simultaneously flippable edges. Such edges are related to the notion of convex decompositions spanned by S.

We prove a worst-case tight lower bound for the number of pseudo-simultaneously flippable edges in a triangulation in terms of the number of vertices. We use this bound for deriving new upper bounds for the maximal number of crossing-free straight-edge graphs that can be embedded on any fixed set of N points in the plane. We obtain new upper bounds for the number of spanning trees and forests as well. Specifically, let \(\mathsf{tr}(N)\) denote the maximum number of triangulations on a set of N points in the plane. Then we show [using the known bound \(\mathsf{tr}(N) < 3{0}^{N}\)] that any N-element point set admits at most \(6.928{3}^{N} \cdot \mathsf{tr}(N) < 207.8{5}^{N}\) crossing-free straight-edge graphs, \(O(4.702{2}^{N}) \cdot \mathsf{tr}(N) = O(141.0{7}^{N})\) spanning trees, and \(O(5.351{4}^{N}) \cdot \mathsf{tr}(N) = O(160.5{5}^{N})\) forests. We also obtain upper bounds for the number of crossing-free straight-edge graphs that have cN, fewer than cN, or more than cN edges, for any constant parameter c, in terms of c and N.

Keywords

Convex Hull Span Tree Planar Graph Convex Polygon Interior Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Work on this chapter by Micha Sharir and Adam Sheffer was partially supported by Grant 338/09 from the Israel Science Fund. Work by Micha Sharir was also supported by NSF Grant CCF-08-30272, by Grant 2006/194 from the U.S.-Israel Binational Science Foundation, and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. Work by Csaba D. Tóth was supported in part by NSERC Grant RGPIN 35586. Research by this author was conducted at ETH Zürich. Emo Welzl acknowledges support from the EuroCores/EuroGiga/ComPoSe SNF Grant 20GG21_134318/1. Part of the work on this chapter was done at the Centre Interfacultaire Bernoulli (CIB), EPFL, Lausanne, during the Special Semester on Discrete and Computational Geometry, Fall 2010, and was supported by the Swiss National Science Foundation.

References

  1. 1.
    O. Aichholzer, T. Hackl, C. Huemer, F. Hurtado, H. Krasser, B. Vogtenhuber, On the number of plane geometric graphs. Graphs Comb. 23(1), 67–84 (2007)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    M. Ajtai, V. Chvátal, M.M. Newborn, E. Szemerédi, Crossing-free subgraphs. Ann. Discr. Math. 12, 9–12 (1982)MATHGoogle Scholar
  3. 3.
    P. Bose, F. Hurtado, Flips in planar graphs. Comput. Geom. Theor. Appl. 42(1), 60–80 (2009)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    P. Bose, G. Toussaint, No quadrangulation is extremely odd, in Algorithms and Computations. Lecture Notes in Computer Science, vol. 1004 (Springer-Verlag, Berlin, 1995), pp. 372–381Google Scholar
  5. 5.
    K. Buchin, C. Knauer, K. Kriegel, A. Schulz, R. Seidel, On the number of cycles in planar graphs, in Proceedings of the 17th Computing and Combinatorics Conference. Lecture Notes Computer Science, vol. 4598 (Springer, Berlin, 2007), pp. 97–107Google Scholar
  6. 6.
    K. Buchin, A. Schulz, On the number of spanning trees a planar graph can have, in Proceedings of the 18th Annual European Symposium on Algorithms. Lecture Notes Computer Science, vol. 6346 (Springer, Berlin, 2010), pp. 110–121Google Scholar
  7. 7.
    J.A. De Loera, J. Rambau, F. Santos, Triangulations: Structures for Algorithms and Applications (Springer, Berlin, 2010)MATHGoogle Scholar
  8. 8.
    M.O. Denny, C.A. Sohler, Encoding a triangulation as a permutation of its point set, in Proceedings of the 9th Canadian Conference on Computational Geometry, 1997, Kingston, Ontario, Canada, pp. 39–43Google Scholar
  9. 9.
    A. Dumitrescu, A. Schulz, A. Sheffer, Cs.D. Tóth, Bounds on the maximum multiplicity of some common geometric graphs, in Proceedings of the 28th Symposium on Theoretical Aspects of Computer Science, 2011, vol. 5 of LIPICS, Schloss Dagstuhl, Dagstuhl, pp. 637–648Google Scholar
  10. 10.
    I. Fáry, On straight line representations of planar graphs. Acta Sci. Math. (Szeged) 11, 229–233 (1948)MATHGoogle Scholar
  11. 11.
    P. Flajolet, M. Noy, Analytic combinatorics of non-crossing configurations. Discr. Math. 204, 203–229 (1999)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    S. Fortune, Voronoi diagrams and Delaunay triangulations, in Computing in Euclidean Geometry, ed. by D.A. Du, F.K. Hwang (World Scientific, New York, 1992), pp. 193–233CrossRefGoogle Scholar
  13. 13.
    J. Galtier, F. Hurtado, M. Noy, S. Pérennes, J. Urrutia, Simultaneous edge flipping in triangulations. Int. J. Comput. Geom. Appl. 13(2), 113–133 (2003)MATHCrossRefGoogle Scholar
  14. 14.
    A. García, M. Noy, J. Tejel, Lower bounds on the number of crossing-free subgraphs of K N. Comput. Geom. Theor. Appl. 16(4), 211–221 (2000)MATHCrossRefGoogle Scholar
  15. 15.
    J. García-Lopez, M. Nicolás, Planar point sets with large minimum convex partitions. Abstracts 22nd European Workshop on Computational Geometry, 2006, Delphi, Greece, pp. 51–54Google Scholar
  16. 16.
    O. Giménez, M. Noy, Asymptotic enumeration and limit laws of planar graphs. J. Am. Math. Soc. 22, 309–329 (2009)MATHCrossRefGoogle Scholar
  17. 17.
    Ø. Hjelle, M. Dæhlen, Triangulations and Applications (Springer, Berlin, 2009)Google Scholar
  18. 18.
    K. Hosono, On convex decompositions of a planar point set. Discr. Math. 309, 1714–1717 (2009)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    F. Hurtado, M. Noy, J. Urrutia, Flipping edges in triangulations. Discr. Comput. Geom. 22, 333–346 (1999)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    L. Lovász, M. Plummer, Matching Theory (North Holland, Budapest, 1986)MATHGoogle Scholar
  21. 21.
    R.C. Mullin, On counting rooted triangular maps. Can. J. Math. 7, 373–382 (1965)MathSciNetCrossRefGoogle Scholar
  22. 22.
    A. Razen, J. Snoeyink, E. Welzl, Number of crossing-free geometric graphs vs. triangulations. Electron. Notes Discr. Math. 31, 195–200 (2008)Google Scholar
  23. 23.
    A. Ribó Mor, Realizations and counting problems for planar structures: trees and linkages, polytopes and polyominos. Ph.D. thesis, Freie Universität, Berlin, 2005Google Scholar
  24. 24.
    G. Rote, The number of spanning trees in a planar graph. Oberwolfach Rep. 2, 969–973 (2005)Google Scholar
  25. 25.
    F. Santos, R. Seidel, A better upper bound on the number of triangulations of a planar point set. J. Comb. Theor. Ser. A 102(1), 186–193 (2003)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    M. Sharir, A. Sheffer, Counting triangulations of planar point sets. Electron. J. Comb. 18(1), P70 (2011)MathSciNetGoogle Scholar
  27. 27.
    M. Sharir, A. Sheffer, E. Welzl, On degrees in random triangulations of point sets. J. Comb. Theor. A 118, 1979–1999 (2011)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    M. Sharir, A. Sheffer, E. Welzl, Counting plane graphs: perfect matchings, spanning cycles, and Kasteleyn’s technique. Proc. 28th ACM Symp. on Computational Geometry, 2012, ACM, New York, pp. 189–198Google Scholar
  29. 29.
    M. Sharir, E. Welzl, Random triangulations of planar point sets, in Proceedings of the 22nd ACM Symposium on Computational Geometry, 2006, ACM, New York, pp. 273–281Google Scholar
  30. 30.
    D.L. Souvaine, C.D. Tóth, A. Winslow, Simultaneously flippable edges in triangulations, in Proceedings of the XIV Spanish Meeting on Computational Geometry, 2011, Alcalá de Henares, Spain, pp. 137–140Google Scholar
  31. 31.
    R.P. Stanley, Enumerative Combinatorics, vol. 2 (Cambridge University Press, Cambridge, 1999)CrossRefGoogle Scholar
  32. 32.
    W.T. Tutte, A census of planar maps. Can. J. Math. 15, 249–271 (1963)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    J. Urrutia, Open problem session, in Proceedings of the 10th Canadian Conference on Computational Geometry, McGill University, Montréal, 1998Google Scholar
  34. 34.
    K. Wagner, Bemerkungen zum Vierfarbenproblem. J. Deutsch. Math.-Verein. 46, 26–32 (1936)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Michael Hoffmann
    • 1
  • André Schulz
    • 2
  • Micha Sharir
    • 3
    • 4
  • Adam Sheffer
    • 3
  • Csaba D. Tóth
    • 5
  • Emo Welzl
    • 1
  1. 1.Institute of Theoretical Computer ScienceETH ZürichZürichSwitzerland
  2. 2.Institut für Mathematische Logik und GrundlagenforschungUniversität MünsterMünsterGermany
  3. 3.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  4. 4.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  5. 5.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

Personalised recommendations