Graphs and Combinatorics

, Volume 32, Issue 3, pp 923–942 | Cite as

Counting Carambolas

  • Adrian Dumitrescu
  • Maarten Löffler
  • André Schulz
  • Csaba D. Tóth
Original Paper

Abstract

We give upper and lower bounds on the maximum and minimum number of geometric configurations of various kinds present (as subgraphs) in a triangulation of n points in the plane. Configurations of interest include convex polygons, star-shaped polygons and monotone paths. We also consider related problems for directed planar straight-line graphs.

Keywords

Convex polygon Star-shaped polygon Monotone path  Plane graph Triangulation Counting 

References

  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)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ajtai, M., Chvátal, V., Newborn, M., Szemerédi, E.: Crossing-free subgraphs. Ann. Discrete Math. 12, 9–12 (1982)MathSciNetMATHGoogle Scholar
  3. 3.
    Buchin, K., Knauer, C., Kriegel, K., Schulz, A., Seidel, R.: On the number of cycles in planar graphs. In: Proceedings of 13th Annual International Conference on Computing and Combinatorics (COCOON), LNCS 4598, pp. 97–107. Springer, Berlin (2007)Google Scholar
  4. 4.
    Buchin, K., Schulz, A.: On the number of spanning trees a planar graph can have. In: Proc. 18th Annual European Symposium on Algorithms (ESA), LNCS 6346, pp. 110–121. Springer, Berlin (2010)Google Scholar
  5. 5.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Berlin (2008)CrossRefMATHGoogle Scholar
  6. 6.
    Dumitrescu, A., Rote, G., Tóth, Cs. D.: Monotone paths in planar convex subdivisions. In: Bezdek, K., Deza, A., Ye, Y. (eds) Discrete Geometry and Optimization, Fields Institute Communications, vol. 69, pp. 79–104. Springer, Berlin (2013)Google Scholar
  7. 7.
    Dumitrescu, A., Schulz, A., Sheffer, A., Tóth, CsD: Bounds on the maximum multiplicity of some common geometric graphs. SIAM J. Discrete Math. 27(2), 802–826 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dumitrescu, A., Tóth, Cs D.: Computational geometry column 54. SIGACT News Bull. 43(4), 90–97 (2012)CrossRefGoogle Scholar
  9. 9.
    Dumitrescu, A., Tóth, Cs. D.: Convex polygons in geometric triangulations. In: Proc. 14th International Symposium on Algorithms and Data Structures (WADS), LNCS 9214, pp. 289–300. Springer, Berlin (2015)Google Scholar
  10. 10.
    Hoffmann, M., Schulz, A., Sharir, M., Sheffer, A., Tóth, C.D., Welzl, E.: Counting plane graphs: flippability and its applications. In: Pach, J. (ed). Thirty Essays on Geometric Graph Theory, pp. 303–326. Springer, Berlin (2013)Google Scholar
  11. 11.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)CrossRefMATHGoogle Scholar
  12. 12.
    Hurtado, F., Noy, M., Urrutia, J.: Flipping edges in triangulations. Discrete Comput. Geom. 22(3), 333–346 (1999)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Löffler, M., Schulz, A., Tóth, Cs.D.: Counting carambolas. In: Proc. 25th Canadian Conference on Computational Geometry (CCCG), pp. 163–168. Waterloo (2013)Google Scholar
  14. 14.
    Pach, J., Tóth, G.: Monotone drawings of planar graphs. J. Graph Theory 46(1), 39–47 (2004)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    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)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    van Kreveld, M., Löffler, M., Pach, J.: How many potatoes are in a mesh? In: Proc. 23rd International Symposium on Algorithms and Computation (ISAAC), LNCS 7676, pp. 166–176. Springer, Berlin (2012)Google Scholar

Copyright information

© Springer Japan 2015

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Maarten Löffler
    • 2
  • André Schulz
    • 3
  • Csaba D. Tóth
    • 4
    • 5
  1. 1.Department of Computer ScienceUniversity of Wisconsin-MilwaukeeMilwaukeeUSA
  2. 2.Department of Computing and Information SciencesUtrecht UniversityUtrechtThe Netherlands
  3. 3.LG Theoretische InformatikFernUniversität HagenHagenGermany
  4. 4.Department of MathematicsCalifornia State University NorthridgeLos AngelesUSA
  5. 5.Department of Computer ScienceTufts UniversityMedfordUSA

Personalised recommendations