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Convex Polygons in Geometric Triangulations

  • ADRIAN DUMITRESCU (a1) and CSABA D. TÓTH (a2) (a3)

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 Ω(1.5028 n ) due to the same authors. Given a planar straight-line graph G with n vertices, we also show how to compute efficiently the number of convex polygons in G.

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An extended abstract of this paper appeared in the Proceedings of the 29th International Symposium on Algorithms and Data Structures (WADS 2015), Vol. 9214 of Lecture Notes in Computer Science, Springer, 2015, pp. 289–300.

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[1] Aichholzer, O., Hackl, T., Vogtenhuber, B., Huemer, C. Hurtado, F. and Krasser, H. (2007) On the number of plane geometric graphs. Graphs Combin. 23 6784.
[2] Ajtai, M., Chvátal, V., Newborn, M. and Szemerédi, E. (1982) Crossing-free subgraphs. Ann. Discrete Math. 12 912.
[3] Alvarez, V., Bringmann, K., Ray, S. and Seidel, R. (2015) Counting triangulations and other crossing-free structures approximately. Comput. Geom. Theory Appl. 48 386397.
[4] Alvarez, V. and Seidel, R. (2013) A simple aggregative algorithm for counting triangulations of planar point sets and related problems. In Proc. 29th Symposium on Computational Geometry (SoCG), ACM Press, pp. 1–8.
[5] Aronov, B., van Kreveld, M., Löffler, M. and Silveira, R. I. (2011) Peeling meshed potatoes. Algorithmica 60 349367.
[6] Buchin, K., Knauer, C., Kriegel, K., Schulz, A. and Seidel, R. (2007) On the number of cycles in planar graphs. In Proc. 13th Annual International Conference on Computing and Combinatorics (COCOON), Vol. 4598 of Lecture Notes in Computer Science, Springer, pp. 97–107.
[7] Chang, J. S. and Yap, C. K. (1986) A polynomial solution for the potato-peeling problem. Discrete Comput. Geom. 1 155182.
[8] Cormen, T. H., Leiserson, C. E., Rivest, R. L. and Stein, C. (2009) Introduction to Algorithms, third edition, MIT Press.
[9] Dumitrescu, A., Löffler, M., Schulz, A. and Tóth, C. D. (2016) Counting carambolas. Graphs Combin. 32 923942.
[10] Dumitrescu, A., Mandal, R. and Tóth, C. D. (2016) Monotone paths in geometric triangulations. arXiv:1608.04812. Extended abstract of an earlier version in Proc. 27th International Workshop on Combinatorial Algorithms (IWOCA 2016), Vol. 9843 of Lecture Notes in Computer Science, Springer, pp. 411–422.
[11] Dumitrescu, A., Schulz, A., Sheffer, A. and Tóth, C. D. (2013) Bounds on the maximum multiplicity of some common geometric graphs. SIAM J. Discrete Math. 27 802826.
[12] Dumitrescu, A. and Tóth, C. D. (2012) Computational Geometry Column 54. SIGACT News Bull. 43 9097.
[13] Dumitrescu, A. and Tóth, C. D. (2017) Convex polygons in geometric triangulations. arXiv:1411.1303v3
[14] Eppstein, D., Overmars, M., Rote, G. and Woeginger, G. (1992) Finding minimum area k-gons. Discrete Comput. Geom. 7 4558.
[15] Erdős, P. (1978) Some more problems on elementary geometry. Gazette Austral. Math. Soc. 5 5254.
[16] Erdős, P. and Szekeres, G. (1935) A combinatorial problem in geometry. Compositio Math. 2 463470.
[17] García, A., Noy, M. and Tejel, A. (2000) Lower bounds on the number of crossing-free subgraphs of K N . Comput. Geom. Theory Appl. 16 211221.
[18] Goodman, J. E. (1981) On the largest convex polygon contained in a non-convex n-gon, or how to peel a potato. Geometria Dedicata 11 99106.
[19] van Kreveld, M. J., Löffler, M. and Pach, J. (2012) How many potatoes are in a mesh? In Proc. 23rd International Symposium on Algebraic Computation (ISAAC), Vol. 7676 of Lecture Notes in Computer Science, Springer, pp. 166–176.
[20] Marx, D. and Miltzow, T. (2016) Peeling and nibbling the cactus: Subexponential-time algorithms for counting triangulations and related problems. In Proc. 32nd International Symposium on Computational Geometry (SoCG), LIPIcs 51, Schloss Dagstuhl, article 52.
[21] Morris, W. and Soltan, V. (2000) The Erdős–Szekeres problem on points in convex position: A survey. Bull. Amer. Math. Soc. 37 437458.
[22] Razen, A., Snoeyink, J. and Welzl, E. (2008) Number of crossing-free geometric graphs vs. triangulations. Electron. Notes Discrete Math. 31 195200.
[23] Razen, A. and Welzl, E. (2011) Counting plane graphs with exponential speed-up. In Rainbow of Computer Science Calude, C. S. et al., eds), Springer, pp. 3646.
[24] Sharir, M. and Sheffer, A. (2011) Counting triangulations of planar point sets. Electron. J. Combin. 18 P70.
[25] Sharir, M. and Sheffer, A. (2013) Counting plane graphs: Cross-graph charging schemes. Combin. Probab. Comput. 22 935954.
[26] Sharir, M., Sheffer, A. and Welzl, E. (2013) Counting plane graphs: Perfect matchings, spanning cycles, and Kasteleyn's technique. J. Combin. Theory Ser. A 120 777794.
[27] Sharir, M. and Welzl, E. (2006) On the number of crossing-free matchings, cycles, and partitions. SIAM J. Comput. 36 695720.
[28] Sheffer, A. (2015) Numbers of plane graphs.
[29] Wettstein, M. (2016) Counting and enumerating crossing-free geometric graphs. In Proc. 30th Symposium on Computational Geometry (SOCG), ACM Press, pp. 1–10. Full paper available at arXiv:1604.05350.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
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