Advertisement

Plane Geometric Graph Augmentation: A Generic Perspective

  • Ferran Hurtado
  • Csaba D. Tóth

Abstract

Graph augmentation problems are motivated by network design and have been studied extensively in optimization. We consider augmentation problems over plane geometric graphs, that is, graphs given with a crossing-free straight-line embedding in the plane. The geometric constraints on the possible new edges render some of the simplest augmentation problems intractable, and in many cases only extremal results are known. We survey recent results, highlight common trends, and gather numerous conjectures and open problems.

Keywords

Planar Graph Hamiltonian Cycle Geometric Graph Congestion Game Stretch Factor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Research by Ferran Hurtado has been partially supported by projects MEC MTM2009-07242, ESF EUROCORES programme EuroGIGA, CRP ComPoSe: MICINN Project EUI-EURC-2011-4306, and Gen. Catalunya DGR 2009SGR1140. Research by Csaba Tóth has been partially supported by NSERC Grant RGPIN 35586 and the NSF Grant CCF-0830734, by the Tóth has been conducted at Tufts University, Medford, MA, USA.

References

  1. 1.
    M. Abellanas, A. García, F. Hurtado, J. Tejel, J. Urrutia, Augmenting the connectivity of geometric graphs. Comput. Geom. Theor. Appl. 40(3), 220–230 (2008)zbMATHCrossRefGoogle Scholar
  2. 2.
    O. Aichholzer, S. Bereg, A. Dumitrescu, A. García, C. Huemer, F. Hurtado, M. Kano, A. Márquez, D. Rappaport, S. Smorodinsky, D. Souvaine, J. Urrutia, D. Wood, Compatible geometric matchings. Comput. Geom. Theor. Appl. 42, 617–626 (2009)zbMATHCrossRefGoogle Scholar
  3. 3.
    O. Aichholzer, D. Bremner, E.D. Demaine, F. Hurtado, E. Kranakis, H. Krasser, S. Ramaswami, S. Sethia, J. Urrutia, Games on triangulations. Theor. Comp. Sci. 343(1–2), 42–71 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    O. Aichholzer, T. Hackl, M. Hoffmann, C. Huemer, A. Pór, F. Santos, B. Speckmann, B. Vogtenhuber, Maximizing maximal angles for plane straight line graphs, in Proceedings of the 10th Workshop on Algorithms and Data Structures, LNCS, vol. 4619 (Springer, Berlin, 2007), pp. 458–469Google Scholar
  5. 5.
    O. Aichholzer, T. Hackl, C. Huemer, F. Hurtado, H. Krasser, B. Vogtenhuber, On the number of plane geometric graphs. Graphs Combinator. 23(1), 67–84 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    M. Ajtai, V. Chvátal, M.M. Newborn, E. Szemerédi, Crossing-free subgraphs. Ann. Discrete Math. 12, 9–12 (1982)zbMATHGoogle Scholar
  7. 7.
    M. Al-Jubeh, M. Hoffmann, M. Ishaque, D.L. Souvaine, C.D. Tóth, Convex partitions with 2-edge connected dual graphs. J. Combin. Opt., 22(3), 409–425 (2011)zbMATHCrossRefGoogle Scholar
  8. 8.
    M. Al-Jubeh, M. Ishaque, K. Rédei, D.L. Souvaine, C.D. Tóth, Tri-edge-connectivity augmentation for planar straight line graphs. Algorithmica 61(4), 971–999 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    N. Alon, S. Rajagopalan, S. Suri, Long non-crossing configurations in the plane. Fundam. Inform. 22(4), 385–394 (1995)MathSciNetzbMATHGoogle Scholar
  10. 10.
    B. Aronov, K. Buchin, M. Buchin, M. Van Kreveld, M. Löffler, J. Luo, R.I. Silveira, B. Speckmann, Connect the dot: computing feed-links with minimum dilation, in Proceedings of the Algorithms Data Structures Symposium (WADS), LNCS, vol. 5664 (Springer, Berlin, 2009), pp. 49–60Google Scholar
  11. 11.
    T. Asano, S.K. Ghosh, T.C. Shermer, in Visibility in the Plane, ed. by J.R. Sack, J. Urrutia. Handbook on Computational Geometry (Elsevier, Amsterdam, 2000), pp. 829–876Google Scholar
  12. 12.
    F. Aurenhammer, Y. Xu, in Optimal Triangulations, ed. by C.A. Floudas, P.M. Pardalos. Encyclopedia of Optimization, vol. 4 (Kluwer, Dordrecht, 2001), pp. 160–166Google Scholar
  13. 13.
    A. Bagheri, M. Razzazi, Planar straight-line point-set embedding of trees with partial embeddings. Inf. Proc. Lett. 110, 521–523 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    I. Bárány, A. Pór, P. Valtr, Paths with no small angles. SIAM J. Discrete Math. 23(4), 1655–1666 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    P. Bose, On embedding an outer-planar graph in a point set. Comput. Geom. Theor. Appl. 23, 303–312 (2002)zbMATHCrossRefGoogle Scholar
  16. 16.
    P. Bose, J. Gudmundsson, M. Smid, Constructing plane spanners of bounded degree and low weight. Algorithmica 42, 249–264 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    P. Bose, L. Devroye, M. Löffler, J. Snoeyink, V. Verma, The spanning ratio of the Delaunay triangulation is greater than π ∕ 2. Comput. Geom. Theor. Appl. 44(2), 121–127 (2011)zbMATHCrossRefGoogle Scholar
  18. 18.
    P. Bose, M.E. Houle, G.T. Toussaint, Every set of disjoint line segments admits a binary tree. Discrete Comput. Geom. 26(3), 387–410 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    P. Bose, M. McAllister, J. Snoeyink, Optimal algorithms to embed trees in a point set. J. Graph Algorithm. Appl. 1(2), 1–15 (1997)MathSciNetCrossRefGoogle Scholar
  20. 20.
    P. Bose, G.T. Toussaint, Growing a tree from its branches. J. Algorithm. 19, 86–103 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    P. Brass, E. Cenek, C. Duncan, A. Efrat, C. Erten, D. Ismailescu, S. Kobourov, A. Lubiw, J. Mitchell. On simultaneous planar graph embeddings. Comput. Geom. Theor. Appl. 36(2), 117–130 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    P. Brass, W. Moser, J. Pach, Research Problems in Discrete Geometry. (Springer, Berlin, 2005)Google Scholar
  23. 23.
    S. Cabello, Planar embeddability of the vertices of a graph using a fixed point set is NP-hard. J. Graph Algorith. Appl. 10, 353–363 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    I. Caragiannis, C. Kaklamanis, E. Kranakis, D. Krizanc, A. Wiese, Communication in wireless networks with directional antennas, in Proceedings of the 20th ACM Symposium on Parallel Algorithms and Architectures (ACM, New York, 2008), pp. 344–351Google Scholar
  25. 25.
    E. Cheng, T. Jordán, Succesive edge-connectivity augmentation problems. Math. Program. 84, 577–593 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    L.P. Chew, There are planar graphs almost as good as the complete graph. J. Comput. Syst. Sci. 39(2), 205–219 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    M. Chrobak, H. Karloff, A lower bound on the size of universal sets for planar graphs. SIGACT News 20, 83–86 (1989)CrossRefGoogle Scholar
  28. 28.
    M. Damian, R. Flatland, Spanning properties of graphs induced by directional antennas, in Proceedings 20th Fall Workshop on Computational Geometry (Stony Brook, NY, 2010)Google Scholar
  29. 29.
    E.D. Demaine, J.S.B. Mitchell, J. O’Rourke (eds.) The Open Problems Project. http://cs.smith.edu/$^\sim$orourke/TOPP/welcome.html
  30. 30.
    T.K. Dey, M.B. Dillencourt, S.K. Ghosh, J.M. Cahill, Triangulating with high connectivity. Comput. Geom. Theor. Appl. 8, 39–56 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    G. Di Battista, P. Eades, R. Tamassia, I.G. Tollis, Graph Drawing: Algorithms for the Visualization of Graphs (Prentice Hall, Englewood Cliffs, NJ, 1998)Google Scholar
  32. 32.
    S. Dobrev, E. Kranakis, D. Krizanc, O. Morales, J. Opatrny, L. Stacho, Strong connectivity in sensor networks with given number of directional antennae of bounded angle, in Proceedings of the 4th Conference on Combinatorial Optimization and Applications, LNCS, vol. 6509 (Springer, Berlin, 2010), pp. 72–86Google Scholar
  33. 33.
    V. Dujmović, D. Eppstein, M. Suderman, D.R. Wood, Drawings of planar graphs with few slopes and segments. Comput. Geom. Theor. Appl. 38, 194–212 (2007)zbMATHCrossRefGoogle Scholar
  34. 34.
    A. Dumitrescu, J. Pach, G. Tóth, Drawing Hamiltonian cycles with no large angles, in Proceedings of the 17th Symposium on Graph Drawing (GD’09), LNCS, vol. 5849 (Springer, Berlin, 2010), pp. 3–14Google Scholar
  35. 35.
    A. Dumitrescu, A. Schulz, A. Sheffer, C.D. Tóth, Bounds on the maximum multiplicity of some common geometric graphs, in Proceedings of the Symposium on Theorerical Aspects of Computer Science (STACS), LIPICS, Schloss Dagstuhl, vol. 5 of LIPICS (Schloss Dagstuhl, Germany, 2011), pp. 637–648Google Scholar
  36. 36.
    A. Dumitrescu, C.D. Tóth, Long non-crossing configurations in the plane. Discrete Comput. Geom. 44(4), 727–752 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    K.P. Eswaran, R.E. Tarjan, Augmentation problems. SIAM J. Comput. 5(4), 653–665 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    M. Farshi, P. Giannopoulos, J. Gudmundsson, Improving the stretch factor of a geometric network by edge augmentation. SIAM J. Comput. 38(1), 226–240 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    S.P. Fekete, G.J. Woeginger, Angle-restricted tours in the plane. Comput. Geom. Theor. Appl. 8(4), 195–218 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    S. Fialko, T.P. Mutzel, A new approximation algorithm for the planar augmentation problem, in Proceedings of the 9th ACM-SIAM Symposium on Discrete Algorithms, ACM Press. 1998, pp. 260–269Google Scholar
  41. 41.
    A. Frank, Augmenting graphs to meet edge-connectivity requirements. SIAM J. Discrete Math. 5(1), 22–53 (1992)CrossRefGoogle Scholar
  42. 42.
    F. Frati, M. Kaufmann, S. Kobourov, Constrained simultaneous and near-simultaneous embeddings, in Proceedings 15th Symposium on Graph Drawing (GD’07), LNCS, vol. 4875 (Springer, Berlin, 2008), pp. 268–279Google Scholar
  43. 43.
    H. de Fraysseix, J. Pach, R. Pollack, How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    A. García, C. Huemer, F. Hurtado, J. Tejel, P. Valtr, On triconnected and cubic plane graphs on given point sets. Comput. Geom. Theor. Appl. 42(9), 913–922 (2009)zbMATHCrossRefGoogle Scholar
  45. 45.
    A. García, M. Noy, J. Tejel, Lower bounds on the number of crossing-free subgraphs of K N. Comput. Geom. Theor. Appl. 16, 211–221 (2000)zbMATHCrossRefGoogle Scholar
  46. 46.
    J. García-Lopez, M. Nicolás, Planar point sets with large minimum convex partitions, in Abstracts of the 22nd European Workshop on Computational Geometry (Delphi, Greece, 2006), pp. 51–54Google Scholar
  47. 47.
    S.K. Ghosh, Visibility Algorithms in the Plane (Cambridge University Press, Cambridge, 2007)zbMATHCrossRefGoogle Scholar
  48. 48.
    P. Giannopoulos, R. Klein, C. Knauer, et al., Computing geometric minimum-dilation graphs is NP-hard. Int. J. Comput. Geom. Appl. 20(2), 147–173 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    J.E. Goodman, J. O’Rourke (eds.), Handbook of Discrete and Computational Geometry, 2nd edn. (Chapman & Hall and CRC Press, Boca Raton, FL, 2004)zbMATHGoogle Scholar
  50. 50.
    J. Gudmundsson, M. Smid, On spanners of geometric graphs. Int. J. Found. Comp. Sci. 20(1), 135–149 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    C. Gutwenger, P. Mutzel, B. Zey, On the hardness and approximability of planar biconnectivity augmentation, in Proceedings of the 15th Computing and Combinatorics Conference (COCOON), LNCS, vol. 5609 (Springer, Berlin, 2009), pp. 249–257Google Scholar
  52. 52.
    P. Gritzmann, B. Mohar, J. Pach, R. Pollack, Embedding a planar triangulation with vertices at specified points. Am. Math. Monthly 98(2), 165–166 (1991)MathSciNetCrossRefGoogle Scholar
  53. 53.
    M. Hoffmann, B. Speckmann, C.D. Tóth, Pointed binary encompassing trees: simple and optimal. Comput. Geom. Theor. Appl. 43(1), 35–41 (2010)zbMATHCrossRefGoogle Scholar
  54. 54.
    M. Hoffmann, C.D. Tóth, Alternating paths through disjoint line segments. Inf. Proc. Lett. 87(6), 287–294 (2003)zbMATHCrossRefGoogle Scholar
  55. 55.
    M. Hoffmann, C.D. Tóth, Pointed and colored binary encompassing trees, in Proceedings of the 21st Symposium on Computational Geometry (ACM, New York, 2005), pp. 81–90Google Scholar
  56. 56.
    M. Hoffmann, C.D. Tóth, Segment endpoint visibility graphs are Hamiltonian. Comput. Geom. Theor. Appl. 26(1), 47–68 (2003)zbMATHCrossRefGoogle Scholar
  57. 57.
    M. Hoffmann, C.D. Tóth, Vertex-colored encompassing graphs, manuscript (2010) http://math.ucalgary.ca/~cdtoth/colored-encom.pdf
  58. 58.
    K. Hosono, On convex decompositions of a planar point set. Discrete Math. 309, 1714–1717 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    F. Hurtado, M. Noy, Counting triangulations of almost-convex polygons. Ars Combinatoria 45, 169–179 (1997)MathSciNetzbMATHGoogle Scholar
  60. 60.
    T.-S. Hsu, Simpler and faster biconnectivity augmentation. J. Algorithm. 45(1), 55–71 (2002)zbMATHCrossRefGoogle Scholar
  61. 61.
    T.-S. Hsu, V. Ramachandran, On finding a minimum augmentation to biconnect a graph. SIAM J. Comput. 22(5), 889–912 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    F. Hurtado, M. Kano, D. Rappaport, C.D. Tóth, Encompassing colored crossing-free geometric graphs. Comput. Geom. Theor. Appl. 39(1), 14–23 (2008)zbMATHCrossRefGoogle Scholar
  63. 63.
    Y. Ikebe, M.A. Perles, A. Tamura, S. Tokunaga, The rooted tree embedding problem into points in the plane. Discrete Comput. Geom. 11, 51–63 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    M. Ishaque, D.L. Souvaine, C.D. Tóth, Disjoint compatible geometric matchings, in Proceedings of the 27th Symposium on Computational Geometry (Paris, 2011) (ACM, New York, 2011), pp. 125–134Google Scholar
  65. 65.
    B. Jackson, T. Jordán, Independence free graphs and vertex connectivity augmentation. J. Combin. Theor. Ser. B 94, 31–77 (2005)zbMATHCrossRefGoogle Scholar
  66. 66.
    I.A. Kanj, L. Perkovic, G. Xia, On spanners and lightweight spanners of geometric graphs. SIAM J. Comput. 39(6), 2132–2161 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    G. Kant, Hexagonal grid drawings, in Proceedings of the 18th Workshop in Graph-Theoretic Concepts in Computer Science (WG’92), LNCS, vol. 657 (Springer-Verlag, Berlin, 1993), pp. 263–276Google Scholar
  68. 68.
    G. Kant, H.L. Bodlaender, Planar graph augmentation problems, in Proceedings of the 2nd Workshop on Algorithms and Data Structures, vol. 519 of LNCS (Springer-Verlag, Berlin, 1991), pp. 286–298Google Scholar
  69. 69.
    M. Kaufmann, R. Wiese, Embedding vertices at points: few bends suffice for planar graphs. J. Graph Algorithm. Appl. 6(1), 115–129 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    J.M. Keil, C.A. Gutwin, Classes of graphs which approximate the complete Euclidean graph. Discrete Comput. Geom. 7, 13–28 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    E. Kranakis, D. Krizanc, O. Morales, L. Stacho, Bounded length, 2-edge augmentation of geometric planar graphs, in Proceedings of the 4th Conference on Combinatorial Optimization and Applications, LNCS, vol. 6508 (Springer, Berlin, 2010), pp. 385–397Google Scholar
  72. 72.
    M. Kurowski, A 1. 235 lower bound on the number of points needed to draw all n-vertex planar graphs. Inf. Proc. Lett. 92, 95–98 (2004)Google Scholar
  73. 73.
    T. Leighton, New lower bound techniques for VLSI. Math. Syst. Theor. 17, 47–70 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    A. Mirzaian, Hamiltonian triangulations and circumscribing polygons of disjoint line segments. Comput. Geom. Theor. Appl. 2(1), 15–30 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    H. Nagamochi, T. Ibaraki, Augmenting edge-connectivity over the entire range in O(nm) time. J. Algorithm. 30, 253–301 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    G. Narasimhan, M. Smid, Geometric Spanner Networks (Cambridge University Press, Cambridge, 2007)zbMATHCrossRefGoogle Scholar
  77. 77.
    V. Neumann-Lara, E. Rivera-Campo, J. Urrutia, A note on convex decompositions of point sets in the plane. Graphs and Combinatorics 20(2), 223–231 (2004)Google Scholar
  78. 78.
    M. Newborn, W.O.J. Moser, Optimal crossing-free Hamiltonian circuit drawings of K n. J. Combin. Theor. Ser. B 29, 13–26 (1980)MathSciNetCrossRefGoogle Scholar
  79. 79.
    W. Mulzer, G. Rote, Minimum-weight triangulation is NP-hard. J. ACM 55(2), article 11 (2008)Google Scholar
  80. 80.
    J. O’Rourke, Visibility, ed. by J.E. Goodman, J. O’Rourke. Handbook of Discrete and Computational Geometry, 2nd edn. (CRC Press, Boca Raton, FL, 2004), pp. 643–665Google Scholar
  81. 81.
    J. O’Rourke, J. Rippel, Two segment classes with Hamiltonian visibility graphs. Comput. Geom. Theor. Appl. 4, 209–218 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    J. Pach, E. Rivera, Two segment classes with Hamiltonian visibility graphs. Comput. Geom. Theor. Appl. 10, 121–124 (1998)zbMATHCrossRefGoogle Scholar
  83. 83.
    M. Patrignani, On extending a partial straight-line drawing. Int. J. Found. Comput. Sci. 17(5), 1061–1069 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    D. Rappaport, Computing simple circuits from a set of line segments is NP-complete. SIAM J. Comput. 18(6), 1128–1139 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    D. Rappaport, H. Imai, G.T. Toussaint, Computing simple circuits from a set of line segments. Discrete Comput. Geom. 5(3), 289–304 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    A. Rosenthal, A. Goldner, Smallest augmentations to biconnect a graph. SIAM J. Comput. 6, 55–66 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  87. 87.
    G. Rote, F. Santos, I. Streinu, in Pseudo-triangulations—A Survey, ed. by J.E. Goodman, J. Pach. Surveys on Discrete and Computational Geometry: Twenty Years Later, vol 453 of Contemporary Mathematics (AMS, Providence, RI, 2008), pp. 343–410Google Scholar
  88. 88.
    T. Roughgarden, On the severity of Braess’s paradox: Designing networks for selfish users is hard. J. Comput. Syst. Sci. 72(5), 922–953 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  89. 89.
    T. Roughgarden, in Routing Games, ed. by N. Nisan et al. Algorithmic Game Theory,  Chapter 18 (Cambridge University Press, Cambridge, 2007)
  90. 90.
    I. Rutter, A. Wolff, Augmenting the connectivity of planar and geometric graphs. Technical report, Universität Karlsruhe, 2008. Number 2008–3. http://digbib.ubka.uni-karlsruhe.de/volltexte/1000007814
  91. 91.
    J.R. Sack, J. Urrutia (eds.), Handbook of Computational Geometry (North Holland, Amsterdam, 2000)zbMATHGoogle Scholar
  92. 92.
    T. Sakai, J. Urrutia, Convex decompositions of point sets in the plane, in Proceedings of the 7th Japan Conference on Computational Geometry and Graphs, JAIST, 2009Google Scholar
  93. 93.
    W. Schnyder, Embedding planar graphs on the grid, in Proceedings of the 1st ACM-SIAM Symposium on Discrete Algorithms, ACM Press. 1990, pp. 138–148Google Scholar
  94. 94.
    M. Sharir, A. Sheffer, Counting triangulations of planar point sets. Electron. J. Combinat. 18, P70 (2011)MathSciNetGoogle Scholar
  95. 95.
    M. Sharir, A. Sheffer, E. Welzl, Counting plane graphs: perfect matchings, spanning cycles, and Kasteleyn’s technique, in Proceedings of the 28th ACM Symposium on Computational Geometry, (ACM Press, 2012), pp. 189–198.Google Scholar
  96. 96.
    M. Sharir, E. Welzl, On the number of crossing-free matchings (cycles, and partitions). SIAM J. Comput. 36(3), 695–720 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
  98. 98.
    D.L. Souvaine, C.D. Tóth, A vertex-face assignment for plane graphs. Comput. Geom. Theor. Appl. 42(5), 388–394 (2009)zbMATHCrossRefGoogle Scholar
  99. 99.
    I. Streinu, A combinatorial approach to planar non-colliding robot arm motion planning, in Proceedings of the 41st Symposium on Foundations of Computer Science (FoCS) (IEEE, Los Alamitos, CA, 2000), pp. 443–453Google Scholar
  100. 100.
    I. Streinu, Pseudo-triangulations, rigidity and motion planning. Discrete Comput. Geom. 34, 587–635 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    R. Tamassia (ed.), Handbook of Graph Drawing and Visualization CRC Press, 2013Google Scholar
  102. 102.
    E. Tardos, T. Wexler, in Inefficiency of Equilibria in Network Formation Games, ed. by N. Nisan et al. Algorithmic Game Theory,  Chapter 19 (Cambridge University Press, Cambridge, 2007)
  103. 103.
    C.D. Tóth, Alternating paths along axis-parallel segments. Graphs Combinator. 22(4), 527–543 (2006)zbMATHCrossRefGoogle Scholar
  104. 104.
    C.D. Tóth, Connectivity augmentation in planar straight line graphs. Europ. J. of Combinatorics, 33(3), 408–425 (2012)zbMATHCrossRefGoogle Scholar
  105. 105.
    C.D. Tóth, P. Valtr, Augmenting the edge connectivity of planar straight line graphs to three, in Proceedings of the 13th Spanish Meeting on Computational Geomentry, Zaragoza, 2009Google Scholar
  106. 106.
    M. Urabe, M. Watanabe, On a counterexample to a conjecture of Mirzaian. Comput. Geom. Theor. Appl. 2(1), 51–53 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  107. 107.
    J. Urrutia, in Art Gallery and Illumination Problems, ed. by J.R. Sack, J. Urrutia. Handbook on Computational Geometry (Elsevier, Amsterdam, 2000), pp. 973–1127Google Scholar
  108. 108.
    L. Végh, Augmenting undirected node-connectivity by one, in Proceedings of the 42nd Symposium on Theory of Computing (STOC) (ACM, New York, 2010), pp. 563–572Google Scholar
  109. 109.
    T. Watanabe, A. Nakamura, Edge-connectivity augmentation problems. J. Comput. Syst. Sci. 35, 96–144 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  110. 110.
    E. Welzl, The number of triangulations on planar point sets, in Proceedings of the 14th International Symposium (GD’06), LNCS, vol. 4372 (Springer, Berlin, 2007), pp. 1–4Google Scholar
  111. 111.
    C. Wulff-Nilsen, Computing the dilation of edge-augmented graphs in metric spaces. Comput. Geom. Theor. Appl. 43, 68-72 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  112. 112.
    G. Xia, Improved upper bound on the stretch factor of Delaunay triangulations, in Proceedings of the 27th Symposium on Computational Geomentry (SoCG) (ACM, New York, 2011), pp. 264–273Google Scholar
  113. 113.
    G. Xia, L. Zhang, Improved lower bound for the stretch factor of Delaunay triangulations, in Proceedings of the 20th Fall Workshop on Computational Geomentry (Stony Brook, NY, 2010)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada IIUniversitat Politècnica de Catalunya (UPC)BarcelonaSpain
  2. 2.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

Personalised recommendations