Counting Plane Graphs: Cross-Graph Charging Schemes

  • Micha Sharir
  • Adam Sheffer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)

Abstract

We study cross-graph charging schemes for graphs drawn in the plane. These are charging schemes where charge is moved across vertices of different graphs. Such methods have been recently applied to obtain various properties of triangulations that are embedded over a fixed set of points in the plane. We show how this method can be generalized to obtain results for various other types of graphs that are embedded in the plane. Specifically, we obtain a new bound of \(O^*\left(187.53^N \right)\) for the maximum number of crossing-free straight-edge graphs that can be embedded over any specific set of N points in the plane (improving upon the previous best upper bound 207.85N in Hoffmann et al.[14]). We also derive upper bounds for numbers of several other types of plane graphs (such as connected and bi-connected plane graphs), and obtain various bounds on expected vertex-degrees in graphs that are uniformly chosen from the set of all crossing-free straight-edge graphs that can be embedded over a specific point set.

We then show how to apply the cross-graph charging-scheme method for graphs that allow certain types of crossings. Specifically, we consider graphs with no set of k pairwise-crossing edges (more commonly known as k-quasi-planar graphs). For k = 3 and k = 4, we prove that, for any set S of N points in the plane, the number of graphs that have a straight-edge k-quasi-planar embedding over S is only exponential in N.

References

  1. 1.
    Ackerman, E.: On the maximum number of edges in topological graphs with no four pairwise crossing edges. Discrete Comput. Geom. 41(3), 365–375 (2009)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Ackerman, E., Tardos, G.: On the maximum number of edges in quasi-planar graphs. J. Combinat. Theory, Ser. A 114(3), 563–571 (2007)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Aichholzer, O., Hackl, T., Huemer, C., Hurtado, F., Krasser, H., Vogtenhuber, B.: On the number of plane geometric graphs. Graphs and Combinatorics 23(1), 67–84 (2007)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Ajtai, M., Chvátal, V., Newborn, M.M., Szemerédi, E.: Crossing-free subgraphs. Annals Discrete Math. 12, 9–12 (1982)MATHGoogle Scholar
  5. 5.
    Appel, K., Haken, W.: Every planar map is four colorable. Part I. Discharging. Illinois J. Math. 21, 429–490 (1977)MathSciNetMATHGoogle Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    Buchin, K., Schulz, A.: On the Number of Spanning Trees a Planar Graph Can Have. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part I. LNCS, vol. 6346, pp. 110–121. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Dumitrescu, A., Schulz, A., Sheffer, A., Tóth, C.D.: Bounds on the maximum multiplicity of some common geometric graphs. In: Proc. 28th Symp. Theo. Aspects Comp. Sci., pp. 637–648 (2011)Google Scholar
  10. 10.
    Euler, L.: Enumeratio modorum, quibus figurae planae rectilineae per diagonales diuiduntur in triangula. Novi Commentarii Academiae Scientiarum Petropolitanae 7, 13–15 (1761)Google Scholar
  11. 11.
    Flajolet, P., Noy, M.: Analytic combinatorics of non-crossing configurations. Discrete Mathematics 204, 203–229 (1999)MathSciNetMATHCrossRefGoogle Scholar
  12. 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)MATHCrossRefGoogle Scholar
  13. 13.
    Heesch, H.: Untersuchungen zum Vierfarbenproblem. Hochschulscripten 810/a/b, Bibliographisches Institut, Mannheim (1969)Google Scholar
  14. 14.
    Hoffmann, M., Sharir, M., Sheffer, A., Tóth, C.D., Welzl, E.: Counting plane graphs: Flippability and its applications. In: Algorithms and Data Structures Symposium Proc. 12th Algs. and Data Structs. Symp., pp. 524–535 (2011)Google Scholar
  15. 15.
    Lamé, G.: Extrait d’une lettre de M. Lamé à M. Liouville sur cette question: Un polygone convexe ètant donné, de combien de manières peut-on le partager en triangles au moyen de diagonales? Journal de Mathèmatiques Pures et Appliquées 3, 505–507 (1838)Google Scholar
  16. 16.
    Pach, J.: Geometric graph theory. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn., pp. 219–238. CRC Press, Boca Raton (2004)Google Scholar
  17. 17.
    Radoičić, R., Tóth, G.: The discharging method in combinatorial geometry and the Pach-Sharir conjecture. In: Goodman, J.E., Pach, J., Pollack, J. (eds.) Surveys on Discrete and Computational Geometry, pp. 319–342. AMS, Providence (2008)Google Scholar
  18. 18.
    Razen, A., Snoeyink, J., Welzl, E.: Number of crossing-free geometric graphs vs. triangulations. Electronic Notes in Discrete Mathematics 31, 195–200 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Razen, A., Welzl, E.: Counting Plane Graphs with Exponential Speed-Up. In: Calude, C.S., Rozenberg, G., Salomaa, A. (eds.) Rainbow of Computer Science. LNCS, vol. 6570, pp. 36–46. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  20. 20.
    Ribó Mor, A.: Realizations and counting problems for planar structures: Trees and linkages, polytopes and polyominos. Ph.D. thesis, Freie Universität Berlin (2005)Google Scholar
  21. 21.
    Rote, G.: The number of spanning trees in a planar graph. Oberwolfach Reports 2, 969–973 (2005)Google Scholar
  22. 22.
    Santos, F., Seidel, R.: A better upper bound on the number of triangulations of a planar point set. J. Combinat. Theory A 102(1), 186–193 (2003)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Sharir, M., Sheffer, A.: Counting triangulations of planar point sets. Electr. J. Comb. 18(1), P70 (2011)Google Scholar
  24. 24.
    Sharir, M., Sheffer, A., Welzl, E.: On degrees in random triangulations of point sets. J. Combinat. Theory A 118, 1979–1999 (2011)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Sharir, M., Welzl, E.: Random triangulations of planar point sets. In: Proc. 22nd ACM Symp. on Computational Geometry, pp. 273–281 (2006)Google Scholar
  26. 26.
    Valtr, P.: Graph Drawings with no k Pairwise Crossing Edges. In: DiBattista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 205–218. Springer, Heidelberg (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Micha Sharir
    • 1
    • 2
  • Adam Sheffer
    • 1
  1. 1.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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