Convex Polygons in Geometric Triangulations

  • Adrian Dumitrescu
  • Csaba D. Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)


We show that the maximum number of convex polygons in a triangulation of n points in the plane is \(O(1.5029^n)\). This improves an earlier bound of \(O(1.6181^n)\) established by van Kreveld, Löffler, and Pach (2012) and almost matches the current best lower bound of \(\Omega (1.5028^n)\) due to the same authors. We show how to compute efficiently the number of convex polygons in a given a planar straight-line graph with n vertices.


Plane Graph Convex Polygon Dual Graph Geometric Graph 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aichholzer, O., Hackl, T., Vogtenhuber, B., Huemer, C., Hurtado, F., Krasser, H.: On the number of plane geometric graphs. Graphs Combin. 23(1), 67–84 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ajtai, M., Chvátal, V., Newborn, M., Szemerédi, E.: Crossing-free subgraphs. Ann. Discrete Math. 12, 9–12 (1982)zbMATHGoogle Scholar
  3. 3.
    Alvarez, V., Bringmann, K., Curticapean, R., Ray, S.: Counting crossing-free structures. In: Proc. 28th Sympos. on Comput. Geom. (SOCG), pp. 61–68. ACM Press (2012). arXiv:1312.4628
  4. 4.
    Alvarez, V., Seidel, R.: A simple aggregative algorithm for counting triangulations of planar point sets and related problems. In: Proc. 29th Sympos. on Comput. Geom. (SOCG), pp. 1–8. ACM Press (2013)Google Scholar
  5. 5.
    Alvarez, V., Bringmann, K., Ray, S., Ray, S.: Counting triangulations and other crossing-free structures approximately. Comput. Geom. 48(5), 386–397 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry, 3rd edn. Springer (2008)Google Scholar
  7. 7.
    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
  8. 8.
    Dumitrescu, A., Schulz, A., Sheffer, A., Tóth, C.D.: Bounds on the maximum multiplicity of some common geometric graphs. SIAM J. Discrete Math. 27(2), 802–826 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dumitrescu, A., Tóth, C.D.: Computational Geometry Column 54. SIGACT News Bulletin 43(4), 90–97 (2012)CrossRefGoogle Scholar
  10. 10.
    Dumitrescu, A., Tóth, C.D.: Convex polygons in geometric triangulations, November 2014. arXiv:1411.1303
  11. 11.
    Eppstein, D., Overmars, M., Rote, G., Woeginger, G.: Finding minimum area \(k\)-gons. Discrete Comput. Geom. 7(1), 45–58 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Erdős, P.: Some more problems on elementary geometry. Austral. Math. Soc. Gaz. 5, 52–54 (1978)MathSciNetGoogle Scholar
  13. 13.
    Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935)Google Scholar
  14. 14.
    García, A., Noy, M., Tejel, A.: Lower bounds on the number of crossing-free subgraphs of \(K_N\). Comput. Geom. 16(4), 211–221 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    van Kreveld, M., Löffler, M., Pach, J.: How Many potatoes are in a mesh? In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 166–176. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  16. 16.
    Löffler, M., Schulz, A., Tóth, C.D.: Counting carambolas. In: Proc. 25th Canadian Conf. on Comput. Geom. (CCCG), Waterloo, ON, pp. 163–168 (2013)Google Scholar
  17. 17.
    Morris, W., Soltan, V.: The Erdős-Szekeres problem on points in convex position–a survey. Bull. AMS 37, 437–458 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Razen, A., Snoeyink, J., Welzl, E.: Number of crossing-free geometric graphs vs. triangulations, Electron. Notes. Discrete Math. 31, 195–200 (2008)MathSciNetGoogle 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.
    Sharir, M., Sheffer, A.: Counting triangulations of planar point sets. Electron. J. Combin. 18, P70 (2011)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Sharir, M., Sheffer, A.: Counting plane graphs: cross-graph charging schemes. Combin., Probab. Comput. 22(6), 935–954 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sharir, M., Sheffer, A., Welzl, E.: Counting plane graphs: perfect matchings, spanning cycles, and Kasteleyn’s technique. J. Combin. Theory, Ser. A 120(4), 777–794 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Sharir, M., Welzl, E.: On the number of crossing-free matchings, cycles, and partitions. SIAM J. Comput. 36(3), 695–720 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sheffer, A.: Numbers of plane graphs (version of May, 2015).
  25. 25.
    Wettstein, M.: Counting and enumerating crossing-free geometric graphs. In: Proc. 30th Sympos. on Comput. Geom. (SOCG), pp. 1–10. ACM Press (2014)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.University of Wisconsin–MilwaukeeMilwaukeeUSA
  2. 2.California State University NorthridgeLos AngelesUSA
  3. 3.Tufts UniversityMedfordUSA

Personalised recommendations