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Convex Polygons in Geometric Triangulations

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Book cover Algorithms and Data Structures (WADS 2015)

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Abstract

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.

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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)

    MATH  Google Scholar 

  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. 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. Alvarez, V., Bringmann, K., Ray, S., Ray, S.: Counting triangulations and other crossing-free structures approximately. Comput. Geom. 48(5), 386–397 (2015)

    Article  MathSciNet  Google Scholar 

  6. de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry, 3rd edn. Springer (2008)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dumitrescu, A., Tóth, C.D.: Computational Geometry Column 54. SIGACT News Bulletin 43(4), 90–97 (2012)

    Article  Google Scholar 

  10. Dumitrescu, A., Tóth, C.D.: Convex polygons in geometric triangulations, November 2014. arXiv:1411.1303

  11. Eppstein, D., Overmars, M., Rote, G., Woeginger, G.: Finding minimum area \(k\)-gons. Discrete Comput. Geom. 7(1), 45–58 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Erdős, P.: Some more problems on elementary geometry. Austral. Math. Soc. Gaz. 5, 52–54 (1978)

    MathSciNet  Google Scholar 

  13. Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Morris, W., Soltan, V.: The Erdős-Szekeres problem on points in convex position–a survey. Bull. AMS 37, 437–458 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  20. Sharir, M., Sheffer, A.: Counting triangulations of planar point sets. Electron. J. Combin. 18, P70 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Sharir, M., Sheffer, A.: Counting plane graphs: cross-graph charging schemes. Combin., Probab. Comput. 22(6), 935–954 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  23. Sharir, M., Welzl, E.: On the number of crossing-free matchings, cycles, and partitions. SIAM J. Comput. 36(3), 695–720 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sheffer, A.: Numbers of plane graphs (version of May, 2015). https://adamsheffer.wordpress.com/numbers-of-plane-graphs/

  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 

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Authors and Affiliations

  1. University of Wisconsin–Milwaukee, Milwaukee, USA

    Adrian Dumitrescu

  2. California State University Northridge, Los Angeles, CA, USA

    Csaba D. Tóth

  3. Tufts University, Medford, MA, USA

    Csaba D. Tóth

Authors
  1. Adrian Dumitrescu
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  2. Csaba D. Tóth
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Corresponding author

Correspondence to Csaba D. Tóth .

Editor information

Editors and Affiliations

  1. Carleton University, Ottawa, Canada

    Frank Dehne

  2. Carleton University, Ottawa, Canada

    Jörg-Rüdiger Sack

  3. University of Victoria, Victoria, Canada

    Ulrike Stege

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Dumitrescu, A., Tóth, C.D. (2015). Convex Polygons in Geometric Triangulations. In: Dehne, F., Sack, JR., Stege, U. (eds) Algorithms and Data Structures. WADS 2015. Lecture Notes in Computer Science(), vol 9214. Springer, Cham. https://doi.org/10.1007/978-3-319-21840-3_24

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  • DOI: https://doi.org/10.1007/978-3-319-21840-3_24

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21839-7

  • Online ISBN: 978-3-319-21840-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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