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The Mathematics of Ferran Hurtado: A Brief Survey

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Discrete and Computational Geometry and Graphs (JCDCGG 2015)

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Abstract

In this paper, dedicated to our dear friend Prof. Ferran Hurtado, we survey some of the research and open problems arising from his work in collaboration with his many friends and colleagues.

J. Urrutia—Research supported in part by SEP-CONACYT of México, Proyecto 80268.

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References

  1. Abellanas, M., García, A., Hurtado, F., Tejel, J.: Caminos alternantes. In: Proceedings of the X Encuentros de Geometría Computacional: Sevilla, Junio 16–17, pp. 7–12 (2003)

    Google Scholar 

  2. Abellanas, M., García, A., Hurtado, F., Tejel, J., Urrutia, J.: Augmenting the connectivity of geometric graphs. Comput. Geom. 40(3), 220–230 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abellanas, M., García, J., Hernández, G., Hurtado, F., Serra, O., Urrutia, J.: Updating polygonizations. Comput. Graph. Forum 12(3), 143–152 (1993)

    Article  Google Scholar 

  4. Abellanas, M., García, J., Hernández, G., Noy, M., Ramos, P.A.: Bipartite embeddings of trees in the plane. Discrete Appl. Math. 93(2), 141–148 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Abellanas, M., Hurtado, F., Ramos, P.A.: Tolerance of geometric structures. In: CCCG, pp. 250–255 (1994)

    Google Scholar 

  6. Abellanas, M., Hurtado, F., Ramos, P.A.: Tolerancia de arreglos de segmentos. In: VI Encuentros de Geometría Computacional: Barcelona, 5-6-7 de julio de, Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya: Actas, pp. 77–84. Departament de Matemàtica Aplicada II (1995)

    Google Scholar 

  7. Abrego, B., Dumitrescu, A., Fernández, S., Tóth, C.D.: Computational geometry column 61. ACM SIGACT News 46(2), 65–77 (2015)

    Article  MathSciNet  Google Scholar 

  8. Ackerman, E., Aichholzer, O., Keszegh, B.: Improved upper bounds on the reflexivity of point sets. Comput. Geom. 42(3), 241–249 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ackerman, E., János, P., Pinchasi, R., Rac̆ić, R., Tóth, G.: A note on coloring line arrangements. Electron. J. Combin. 21(2), Paper p2.23 (2012)

    Google Scholar 

  10. Aichholzer, O., Alboul, L.S., Hurtado, F.: On flips in polyhedral surfaces. Int. J. Found. Comput. Sci. 13(02), 303–311 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Aichholzer, O., Aurenhammer, F., Demaine, E.D., Hurtado, F., Ramos, P., Urrutia, J.: On k-convex polygons. Comput. Geom. 45(3), 73–87 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Aichholzer, O., Aurenhammer, F., Hackl, T., Hurtado, F., Pilz, A., Ramos, P., Urrutia, J., Valtr, P., Vogtenhuber, B.: On k-convex point sets. Comput. Geom. 47(8), 809–832 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Aichholzer, O., Aurenhammer, F., Hurtado, F., Krasser, H.: Towards compatible triangulations. Theor. Comput. Sci. 296(1), 3–13 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Aichholzer, O., Fabila-Monroy, R., Flores-Peñaloza, D., Hackl, T., Huemer, C., Urrutia, J.: Empty monochromatic triangles. Comput. Geom. 42(9), 934–938 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Aichholzer, O., Fabila-Monroy, R., Hackl, T., Van Kreveld, M., Pilz, A., Ramos, P., Vogtenhuber, B.: Blocking delaunay triangulations. Comput. Geom. 46(2), 154–159 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Aichholzer, O., Hackl, T., Huemer, C., Hurtado, F., Krasser, H., Vogtenhuber, B.: On the number of plane geometric graphs. Graphs Comb. 23(1), 67–84 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Aichholzer, O., Hackl, T., Huemer, C., Hurtado, F., Vogtenhuber, B.: Large bichromatic point sets admit empty monochromatic 4-gons. SIAM J. Discrete Math. 23(4), 2147–2155 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Aichholzer, O., Hurtado, F., Noy, M.: A lower bound on the number of triangulations of planar point sets. Comput. Geom. 29(2), 135–145 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ajtai, M., Chvátal, V., Newborn, M.M., Szemerédi, E.: Crossing-free subgraphs. North-Holland Math. Stud. 60, 9–12 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  20. Akiyama, J., Urrutia, J.: Simple alternating path problem. Discrete Math. 84(1), 101–103 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Arkin, E.M., Mitchell, J.S.B., Fekete, S.P., Hurtado, F., Noy, M., Sacristán, V., Sethia, S.: On the reflexivity of point sets. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry, pp. 139–156. Springer, New York (2003)

    Chapter  Google Scholar 

  22. Aronov, B., Dulieu, M., Hurtado, F.: Witness (delaunay) graphs. Comput. Geom. 44(6), 329–344 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Aronov, B., Dulieu, M., Hurtado, F.: Witness gabriel graphs. Comput. Geom. 46(7), 894–908 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Aronov, B., Dulieu, M., Hurtado, F.: Witness rectangle graphs. Graphs Comb. 30(4), 827–846 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Bose, P., Cardinal, J., Collette, S., Hurtado, F., Korman, M., Langerman, S., Taslakian, P.: Coloring and guarding arrangements. Discrete Math. Theor. Comput. Sci. 15(3), 139–154 (2013)

    MathSciNet  MATH  Google Scholar 

  26. Bose, P., Czyzowicz, J., Gao, Z., Morin, P., Wood, D.R.: Simultaneous diagonal flips in plane triangulations. J. Graph Theory 54(4), 307–330 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Bose, P., Hurtado, F.: Flips in planar graphs. Comput. Geom. 42(1), 60–80 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Cibulka, J., Kynčl, J., Mészáros, V., Stolař, R., Valtr, P.: Hamiltonian alternating paths on bicolored double-chains. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 181–192. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  29. Claverol, M., Garijo, D., Hurtado, F., Cuevas, D.L., Seara, C.: The alternating path problem revisited. In: Proceedings XIV Spanish Meeting on Computational Geometry, EGC 2011, Alcalá de Henares, Spain, June 27–30, 2011, pp. 115–118. Universidad de Sevilla (2013)

    Google Scholar 

  30. Devillers, O., Hurtado, F., Károlyi, G., Seara, C.: Chromatic variants of the erdos-szekeres theorem on points in convex position. Comput. Geom. 26(3), 193–208 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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 

  32. Enomoto, H., Kanda, K., Masui, T., Oda, Y., Ota, K.: Private communication

    Google Scholar 

  33. Erdös, P.: On some problems of elementary and combinatorial geometry. Annali di Matematica pura ed applicata 103(1), 99–108 (1975)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  35. Frati, F.: Planar packing of diameter-four trees. In: Proceedings of the XXI Canadian Conference on Computational Geometry: Vancouver, BC, August 17–19, 2009, pp. 95–98 (2009)

    Google Scholar 

  36. Frati, F., Geyer, M., Kaufmann, M.: Planar packing of trees and spider trees. Inform. Process. Lett. 109(6), 301–307 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  37. Galtier, J., Hurtado, F., Noy, M., Pérennes, S., Urrutia, J.: Simultaneous edge flipping in triangulations. Int. J. Comput. Geom. Appl. 13(02), 113–133 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  38. García, A., Hernando, C., Hurtado, F., Noy, M., Tejel, J.: Packing trees into planar graphs. J. Graph Theory 40(3), 172–181 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  39. García, A., Huemer, C., Hurtado, F., Tejel, J.: Compatible spanning trees. Comput. Geom. 47(5), 563–584 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. García, A., Hurtado, F., Korman, M., Matos, I., Saumell, M., Silveira, R.I., Tejel, J., Tóth, C.D.: Geometric biplane graphs i: maximal graphs. Graphs Comb. 31(2), 407–425 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  41. García, A., Hurtado, F., Korman, M., Matos, I., Saumell, M., Silveira, R.I., Tejel, J., Tóth, C.D.: Geometric biplane graphs ii: graph augmentation. Graphs Comb. 31(2), 427–452 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. García, A., Noy, M., Tejel, J.: Lower bounds on the number of crossing-free subgraphs of \(k_n\). Comput. Geom. 16(4), 211–221 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  43. García, A., Tejel, J.: Private communication

    Google Scholar 

  44. García-Lopez, J., Nicolás, C.M.: A counterexample about convex partitions. In: IV Jornadas de Matemática Discreta y Algorítmica 2004, Cercedilla, 5–8 Septiembre, 2004, p. 213 (2004)

    Google Scholar 

  45. Geyer, M., Hoffmann, M., Kaufmann, M., Kusters, V., Tóth, C.D.: Planar packing of binary trees. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 353–364. Springer, Heidelberg (2013). doi:10.1007/978-3-642-40104-6_31

    Chapter  Google Scholar 

  46. González-Martínez, A.C., Cravioto-Lagos, J., Urrutia, J.: Almost empty monochromatic polygons in planar point sets. In: Actas XVI Spanish Meeting on Computational Geometry, Barcelona, 1–3 de julio, 2015, pp. 81–84 (2015)

    Google Scholar 

  47. Görlich, A.: Packing cycles and unicyclic graphs into planar graphs. Demonstr. Math. XLI I(4), 673–679 (2009)

    MATH  Google Scholar 

  48. Hajnal, P., Mészáros, V.: Note on noncrossing path in colored convex sets. Discr. Math. Theor. Comput. Sci. (2015, accepted)

    Google Scholar 

  49. Hedetniemi, S.M., Hedetniemi, S.T., Slater, P.J.: A note on packing two trees into \(k_n\). Ars Combin. 11, 149–153 (1981)

    MathSciNet  MATH  Google Scholar 

  50. Horton, J.D.: Sets with no empty convex 7-gons. Can. Math. Bull. 26(4), 482 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  51. Hurtado, F.A.: Problemes geomètrics de visibilitat. Ph.D. thesis, Departamento de Matemática Aplicada i Telemàtica, UPC (1993)

    Google Scholar 

  52. Hurtado, F.: Looking through a window. In: Proceedings of the Fifth Canadian Conference on Computational Geometry: University of Waterloo, August 5–9, 1993, pp. 234–239 (1993)

    Google Scholar 

  53. Hurtado, F., Marc, N., El número de triangulaciones de un polígono. In: Proceedings of the IV Encuentros de Geometría Computacional: Granada, pp. 1–6 (1993)

    Google Scholar 

  54. Hurtado, F., Noy, M.: Ears of triangulations and catalan numbers. Discrete Math. 149(1), 319–324 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  55. Hurtado, F., Noy, M.: Triangulations, visibility graph and reflex vertices of a simple polygon. Comput. Geom. 6(6), 355–369 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  56. Hurtado, F., Noy, M.: Graph of triangulations of a convex polygon and tree of triangulations. Comput. Geom. 13(3), 179–188 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  58. Hurtado, F., Serra, O., Urrutia, J.: Hiding points in arrangements of segments. Discrete Math. 162(1), 187–197 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  59. Kaneko, A., Kano, M., Yoshimoto, K.: Alternating hamilton cycles with minimum number of crossings in the plane. Int. J. Comput. Geom. Appl. 10(01), 73–78 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  60. Kynčl, J., Pach, J., Tóth, G.: Long alternating paths in bicolored point sets. Discrete Math. 308(19), 4315–4321 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  61. Lee, C.W.: The associahedron and triangulations of the \(n\)-gon. Eur. J. Comb. 10(6), 551–560 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  62. Lucas, J.M., Vanbaronaigien, D.R., Ruskey, F.: On rotations and the generation of binary trees. J. Algorithms 15(3), 343–366 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  63. Merino, C., Salazar, G., Urrutia, J.: On the length of longest alternating paths for multicoloured point sets in convex position. Discrete Math. 306(15), 1791–1797 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  64. Oda, Y., Ota, K.: Tight planar packings of two trees. In: Twenty-second European Workshop on Computational Geometry Delphi, Greece March 27–29, 2006, p. 215 (2006)

    Google Scholar 

  65. Pach, J., Tóth, G.: Monochromatic empty triangles in two-colored point sets. Discrete Appl. Math. 161(9), 1259–1261 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  66. Sharir, M., Sheffer, A.: Counting plane graphs: cross-graph charging schemes. Comb. Probab. Comput. 22(06), 935–954 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  67. Sharir, M., Sheffer, A., Welzl, E.: On degrees in random triangulations of point sets. J. Comb. Theory Ser. A 118(7), 1979–1999 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  68. Souvaine, D.L., Tóth, C.D., Winslow, A.: Simultaneously flippable edges in triangulations. In: Márquez, A., Ramos, P., Urrutia, J. (eds.) EGC 2011. LNCS, vol. 7579, pp. 138–145. Springer, Heidelberg (2012). doi:10.1007/978-3-642-34191-5_13

    Chapter  Google Scholar 

  69. Tóth, C.D.: Connectivity augmentation in planar straight line graphs. Eur. J. Comb. 33(3), 408–425 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  70. Urabe, M.: On a partition into convex polygons. Discrete Appl. Math. 64(2), 179–191 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  71. Urabe, M.: On a partition of point sets into convex polygons. In: Proceedings of the Ninth Canadian Conference on Computational Geometry, August 11–13, Queen’s Universiy, Kingston, Ontario, pp. 179–191 (1997)

    Google Scholar 

  72. Urrutia, J.: Coloraciones, tetraedralizaciones, y tetraedros vacíos en coloraciones de conjuntos de puntos en \(\mathbb{R}^3\). In: Proceedings of the X Encuentros de Geometría Computacional: Sevilla, Junio 16–17, 2003, pp. 95–100 (2003)

    Google Scholar 

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Urrutia, J. (2016). The Mathematics of Ferran Hurtado: A Brief Survey. In: Akiyama, J., Ito, H., Sakai, T., Uno, Y. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2015. Lecture Notes in Computer Science(), vol 9943. Springer, Cham. https://doi.org/10.1007/978-3-319-48532-4_25

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