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Counting Carambolas

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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.

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Notes

  1. Throughout this paper, all logarithms are in base 2.

  2. Such paths were inadvertently overlooked in the analysis from [13].

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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)

  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)

  5. de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dumitrescu, A., Tóth, Cs D.: Computational geometry column 54. SIGACT News Bull. 43(4), 90–97 (2012)

    Article  Google Scholar 

  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)

  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)

  11. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    Book  MATH  Google Scholar 

  12. Hurtado, F., Noy, M., Urrutia, J.: Flipping edges in triangulations. Discrete Comput. Geom. 22(3), 333–346 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Löffler, M., Schulz, A., Tóth, Cs.D.: Counting carambolas. In: Proc. 25th Canadian Conference on Computational Geometry (CCCG), pp. 163–168. Waterloo (2013)

  14. Pach, J., Tóth, G.: Monotone drawings of planar graphs. J. Graph Theory 46(1), 39–47 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

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Acknowledgments

A. Dumitrescu was supported in part by NSF grant DMS-1001667. M. Löffler was supported by the Netherlands Organization for Scientific Research (NWO) under grant 639.021.123. Research by Tóth was supported in part by NSERC (RGPIN 35586) and NSF (CCF-1423615). This work was initiated at the workshop “Counting and Enumerating Plane Graphs,” which took place at Schloss Dagstuhl in March, 2013.

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Authors and Affiliations

  1. Department of Computer Science, University of Wisconsin-Milwaukee, Milwaukee, USA

    Adrian Dumitrescu

  2. Department of Computing and Information Sciences, Utrecht University, Utrecht, The Netherlands

    Maarten Löffler

  3. LG Theoretische Informatik, FernUniversität Hagen, Hagen, Germany

    André Schulz

  4. Department of Mathematics, California State University Northridge, Los Angeles, CA, USA

    Csaba D. Tóth

  5. Department of Computer Science, Tufts University, Medford, MA, USA

    Csaba D. Tóth

Authors
  1. Adrian Dumitrescu
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  2. Maarten Löffler
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  3. André Schulz
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  4. Csaba D. Tóth
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Corresponding author

Correspondence to Csaba D. Tóth.

Additional information

A preliminary version of this paper appeared in the Proceedings of the 25th Canadian Conference on Computational Geometry [13].

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Dumitrescu, A., Löffler, M., Schulz, A. et al. Counting Carambolas. Graphs and Combinatorics 32, 923–942 (2016). https://doi.org/10.1007/s00373-015-1621-7

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  • DOI: https://doi.org/10.1007/s00373-015-1621-7

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