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Workshop on Algorithms and Data Structures

WADS 2011: Algorithms and Data Structures pp 524–535Cite as

Counting Plane Graphs: Flippability and Its Applications

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6844))

Abstract

We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane into so called pseudo-simultaneously flippable edges.

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 tr(N) denote the maximum number of triangulations on a set of N points in the plane. Then we show (using the known bound tr(N) < 30N) that any N-element point set admits at most 6.9283N ·tr(N) < 207.85N crossing-free straight-edge graphs, O(4.8795N) ·tr(N) = O(146.39N) spanning trees, and O(5.4723N) ·tr(N) = O(164.17N) forests. We also obtain upper bounds for the number of crossing-free straight-edge graphs that have fewer than cN or more than cN edges, for a constant parameter c, in terms of c and N.

Work 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 paper was done at the Centre Interfacultaire Bernoulli (CIB), during the Special Semester on Discrete and Computational Geometry, Fall 2010, and supported by the Swiss National Science Foundation.

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References

  1. Aichholzer, O., Hackl, T., Huemer, Hurtado, F., Krasser, H., Vogtenhuber, B.: On the number of plane geometric graphs. Graph. Comb. 23(1), 67–84 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ajtai, M., Chvátal, V., Newborn, M.M., Szemerédi, E.: Crossing-free subgraphs. Annals Discrete Math. 12, 9–12 (1982)

    MathSciNet  MATH  Google Scholar 

  3. Bose, P., Hurtado, F.: Flips in planar graphs. Comput. Geom. Theory Appl. 42(1), 60–80 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buchin, K., Knauer, C., Kriegel, K., Schulz, A., Seidel, R.: On the number of cycles in planar graphs. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 97–107. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  5. Buchin, K., Schulz, A.: On the number of spanning trees a planar graph can have. In: de Berg, M., Meyer, U. (eds.) ESA 2010. LNCS, vol. 6346, pp. 110–121. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  6. De Loera, J.A., Rambau, J., Santos, F.: Triangulations: Structures for Algorithms and Applications. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  7. Denny, M.O., Sohler, C.A.: Encoding a triangulation as a permutation of its point set. In: Proc. 9th Canadian Conf. on Computational Geometry, pp. 39–43 (1997)

    Google Scholar 

  8. Dumitrescu, A., Schulz, A., Sheffer, A., Tóth, C.D.: Bounds on the maximum multiplicity of some common geometric graphs. In: Proc. 28th International Symposium on Theoretical Aspects of Computer Science, pp. 637–648 (2011)

    Google Scholar 

  9. Fortune, S.: Voronoi diagrams and Delaunay triangulations. In: Du, D.A., Hwang, F.K. (eds.) Euclidean Geometry and Computers, pp. 193–233. World Scientific Publishing Co., New York (1992)

    Chapter  Google Scholar 

  10. Galtier, J., Hurtado, F., Noy, M., Pérennes, S., Urrutia, J.: Simultaneous edge flipping in triangulations. Internat. J. Comput. Geom. Appl. 13(2), 113–133 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. García-Lopez, J., Nicolás, M.: Planar point sets with large minimum convex partitions. In: Proc. 22nd Euro. Workshop Comput. Geom., Delphi, pp. 51–54 (2006)

    Google Scholar 

  12. García, A., Noy, M., Tejel, J.: Lower bounds on the number of crossing-free subgraphs of K N . Comput. Geom. Theory Appl. 16(4), 211–221 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hjelle, Ø., Dæhlen, M.: Triangulations and Applications. Springer, Berlin (2009)

    Google Scholar 

  14. Hoffmann, M., Sharir, M., Sheffer, A., Tóth, C.D., Welzl, E.: Counting Plane Graphs: Flippability and its Applications, arXiv:1012.0591

    Google Scholar 

  15. Hosono, K.: On convex decompositions of a planar point set. Discrete Math. 309, 1714–1717 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hurtado, F., Noy, M., Urrutia, J.: Flipping edges in triangulations. Discrete Comput. Geom. 22, 333–346 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lovász, L., Plummer, M.: Matching theory. North Holland, Amsterdam (1986)

    MATH  Google Scholar 

  18. Razen, A., Snoeyink, J., Welzl, E.: Number of crossing-free geometric graphs vs. triangulations. Electronic Notes in Discrete Math. 31, 195–200 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ribó, A.: Realizations and Counting Problems for Planar Structures: Trees and Linkages, Polytopes and Polyominos, Ph.D. thesis. Freie Universität Berlin (2005)

    Google Scholar 

  20. Rote, G.: The number of spanning trees in a planar graph. Oberwolfach Reports 2, 969–973 (2005)

    Google Scholar 

  21. Santos, F., Seidel, R.: A better upper bound on the number of triangulations of a planar point set. J. Combinat. Theory Ser. A 102(1), 186–193 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sharir, M., Sheffer, A.: Counting triangulations of planar point sets. Electr. J. Comb. 18(1) (2011), arXiv:0911.3352v2

    Google Scholar 

  23. Sharir, M., Welzl, E.: Random triangulations of planar point sets. In: Proc. 17th Ann. ACM-SIAM Symp. on Discrete Algorithms, pp. 860–869 (2006)

    Google Scholar 

  24. Souvaine, D.L., Tóth, C.D., Winslow, A.: Simultaneously flippable edges in triangulations. In: Proc. XIV Spanish Meeting on Comput. Geom. (to appear, 2011)

    Google Scholar 

  25. Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  26. Urrutia, J.: Open problem session. In: Proc. 10th Canadian Conference on Computational Geometry. McGill University, Montréal (1998)

    Google Scholar 

  27. Wagner, K.: Bemerkungen zum Vierfarbenproblem. J. Deutsch. Math.-Verein. 46, 26–32 (1936)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Theoretical Computer Science, ETH Zürich, CH, 8092, Zürich, Switzerland

    Michael Hoffmann & Emo Welzl

  2. Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel

    Micha Sharir & Adam Sheffer

  3. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Micha Sharir

  4. Department of Mathematics and Statistics, University of Calgary, Calgary, AB, Canada

    Csaba D. Tóth

Authors
  1. Michael Hoffmann
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  2. Micha Sharir
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  3. Adam Sheffer
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  4. Csaba D. Tóth
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  5. Emo Welzl
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Editor information

Editors and Affiliations

  1. School of Computer Science, Parallel Computing and Bioinformatics Laboratory, Carleton University, VISM Building, Room 6210, 1125 Colonel By Drive, K1S 5B6, Ottawa, ON, Canada

    Frank Dehne

  2. Department of Computer Science and Engineering, Polytechnic Institute of New York University, 5 MetroTech Center, 11201, New York, NY, USA

    John Iacono

  3. School of Computer Science, Herzberg Laboratories, Carleton University, Herzberg Building, Room 5350, 1125 Colonel By Drive, K1S 5B6, Ottawa, ON, Canada

    Jörg-Rüdiger Sack

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Hoffmann, M., Sharir, M., Sheffer, A., Tóth, C.D., Welzl, E. (2011). Counting Plane Graphs: Flippability and Its Applications. In: Dehne, F., Iacono, J., Sack, JR. (eds) Algorithms and Data Structures. WADS 2011. Lecture Notes in Computer Science, vol 6844. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22300-6_44

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  • DOI: https://doi.org/10.1007/978-3-642-22300-6_44

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-22299-3

  • Online ISBN: 978-3-642-22300-6

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