Selected Open Problems in Graph Drawing

  • Franz Brandenburg
  • David Eppstein
  • Michael T. Goodrich
  • Stephen Kobourov
  • Giuseppe Liotta
  • Petra Mutzel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2912)


In this manuscript, we present several challenging and interesting open problems in graph drawing. The goal of the listing in this paper is to stimulate future research in graph drawing.


Minimum Span Tree Linear Time Algorithm Simple Polygon Outerplanar Graph Graph Drawing 
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.


  1. 1.
    Abello, J., Kumar, K.: Visibility graphs and oriented matroids. In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol. 894, pp. 147–158. Springer, Heidelberg (1995)Google Scholar
  2. 2.
    Aichholzer, O., Aurenhammer, F., Chen, S.-W., Katoh, N., Taschwer, M., Rote, G., Xu, Y.-F.: Triangulations intersect nicely. Discrete Comput. Geom. 16, 339–359 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Aichholzer, O., Aurenhammer, F., Krasser, H.: On the crossing number of complete graphs. In: Proceedings of the 18th ACM Symposium on Computational Geometry, pp. 19–24. ACM Press, New York (2002)Google Scholar
  4. 4.
    Aronov, B., Erdős, P., Goddard, W., Kleitman, D.J., Klugerman, M., Pach, J., Schulman, L.J.: Crossing families. Combinatorica 14, 127–134 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Barth, W., Jünger, M., Mutzel, P.: Simple and efficient cross counting. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 130–141. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Bertolazzi, P., Di Battista, G., Didimo, W.: Computing orthogonal drawings with the minimum number of bends. In: Rau-Chaplin, A., Dehne, F., Sack, J.-R., Tamassia, R. (eds.) WADS 1997. LNCS, vol. 1272, pp. 331–344. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  7. 7.
    Bhatt, S., Cosmadakis, S.: The complexity of minimizing wire lengths in VLSI layouts. Inform. Process. Lett. 25, 263–267 (1987)zbMATHCrossRefGoogle Scholar
  8. 8.
    Bhatt, S.N., Leighton, F.T.: A framework for solving VLSI graph layout problems. J. Comput. Syst. Sci. 28, 300–343 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Biedl, T.: Drawing outer-planar graphs in O(n log n) area. In: Goodrich, M. (ed.) GD 2002. LNCS, vol. 2528, pp. 54–65. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    Bienstock, D., Dean, N.: Bounds for rectilinear crossing numbers. J. Graph Theory 17, 333 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Blankenship, R., Oporowski, B.: Book embeddings of graphs and minor-closed classes. In: Thirty-Second Southeastern Internat. Conf. Combinatorics, Graph Theory and Computing, Baton Rouge. Dept. of Mathematics, p. 30. Louisiana State University (2001),
  12. 12.
    Bose, P.: On embedding an outer-planar graph in a point set. Comput. Geom. 23(3), 303–312 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Bose, P., Lenhart, W., Liotta, G.: Characterizing proximity trees. Algorithmica 16, 83–110 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Cairns, G., Nikolayevsky, Y.: Bounds for generalized thrackles. Discrete Comput. Geom. 23(2), 191–206 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Calamoneri, T., Sterbini, A.: 3D straight-line grid drawing of 4-colorable graphs. Inform. Process. Lett. 63(2), 97–102 (1997)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Calinescu, G., Fernandes, C.G., Finkler, U., Karloff, H.: A better approximation algorithm for finding planar subgraphs. In: Proc. 7th ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 16–25 (1996)Google Scholar
  17. 17.
    Chan, T.: A near-linear area bound for drawing binary trees. Algorithmica 34, 1–13 (2001)CrossRefGoogle Scholar
  18. 18.
    Chan, T.M., Goodrich, M.T., Kosaraju, S.R., Tamassia, R.: Optimizing area and aspect ratio in straight-line orthogonal tree drawings. In: North, S.C. (ed.) GD 1996. LNCS, vol. 1190, pp. 63–75. Springer, Heidelberg (1997)Google Scholar
  19. 19.
    Chazelle, B.: Reporting and counting segment intersections. J. Comput. Syst. Sci. 32, 156–182 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Chazelle, B.: Triangulating a simple polygon in linear time. Discrete Comput. Geom. 6, 485–524 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Chazelle, B., Incerpi, J.: Triangulation and shape-complexity. ACM Trans. Graph. 3(2), 135–152 (1984)zbMATHCrossRefGoogle Scholar
  22. 22.
    Cheng, S.-W., Xu, Y.-F.: On β-skeleton as a subgraph of the minimum weight triangulation. Theoretical Computer Science 262(1-2), 459–471 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Chrobak, M., Goodrich, M.T., Tamassia, R.: Convex drawings of graphs in two and three dimensions. In: Proc. 12th Annu. ACM Sympos. Comput. Geom., pp. 319–328 (1996)Google Scholar
  24. 24.
    Chrobak, M., Ichi Nakano, S.: Minimum-width grid drawings of plane graphs. Comput. Geom. Theory Appl. 11, 29–54 (1998)zbMATHGoogle Scholar
  25. 25.
    Chrobak, M., Kant, G.: Convex grid drawings of 3-connected planar graphs. Internat. J. Comput. Geom. Appl. 7(3), 211–223 (1997)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Chrobak, M., Karloff, H.: A lower bound on the size of universal sets for planar graphs. SIGACT News 20 (1989)Google Scholar
  27. 27.
    Cohen, R.F., Eades, P., Lin, T., Ruskey, F.: Three-dimensional graph drawing. Algorithmica 17, 199–208 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Dahlhaus, E.: Linear time algorithm to recognize clustered planar graphs and its parallelization. In: Lucchesi, C.L., Moura, A.V. (eds.) LATIN 1998. LNCS, vol. 1380, pp. 239–248. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  29. 29.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Devillers, O., Liotta, G., Preparata, F.P., Tamassia, R.: Checking the convexity of polytopes and the planarity of subdivisions. Comput. Geom. Theory Appl. 11, 187–208 (1998)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice Hall, Upper Saddle River (1999)zbMATHGoogle Scholar
  32. 32.
    Di Battista, G., Garg, A., Liotta, G., Tamassia, R., Tassinari, E., Vargiu, F.: An experimental comparison of four graph drawing algorithms. Comput. Geom. Theory Appl. 7, 303–325 (1997)zbMATHGoogle Scholar
  33. 33.
    Di Battista, G., Liotta, G.: Upward planarity checking: Faces are more than polygons. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 72–86. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  34. 34.
    Di Battista, G., Liotta, G., Lubiw, A., Whitesides, S.: Orthogonal drawings of cycles in 3D space. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 272–283. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  35. 35.
    Di Battista, G., Liotta, G., Lubiw, A., Whitesides, S.: Embedding problems for paths with direction constrained edges. Theoretical Computer Science 289(2), 897–917 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Di Battista, G., Liotta, G., Vargiu, F.: Spirality and optimal orhogonal drawings. SIAM J. Comput. 27(6), 1764–1811 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Di Battista, G., Vismara, L.: Angles of planar triangular graphs. SIAM J. Discrete Math. 9(3), 349–359 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Di Giacomo, E.: Drawing series-parallel graphs on restricted integer 3D grids. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 238–246. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  39. 39.
    Di Giacomo, E., Liotta, G., Meijer, H.: 3D straight-line drawings of k-trees. Technical Report TR-2003-473, Queen’s University, School of Computing (2003)Google Scholar
  40. 40.
    Di Giacomo, E., Liotta, G., Patrignani, M.: Orthogonal 3D shapes of theta graphs. In: Goodrich, M. (ed.) GD 2002. LNCS, vol. 2528, pp. 142–149. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  41. 41.
    Di Giacomo, E., Liotta, G., Wismath, S.: Drawing series-parallel graphs on a box. In: 14th Canadian Conference On Computational Geometry, CCCG 2002 (2002)Google Scholar
  42. 42.
    Di Giacomo, E., Meijer, H.: Track drawings of graphs with constant queue number. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 214–225. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  43. 43.
    Dickerson, M., Keil, J., Montague, M.: A large subgraph of the minimum weight triangulation. Discrete and Computational Geometry 18, 289–304 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Dickerson, M.T., Montague, M.H.: A (usually?) connected subgraph of the minimum weight triangulation. In: Proc. 12th Annu. ACM Sympos. Comput. Geom., pp. 204–213 (1996)Google Scholar
  45. 45.
    Dijdjev, H., Vrťo, I.: An improved lower bound for crossing numbers. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, p. 96. Springer, Heidelberg (2002) (to appear)CrossRefGoogle Scholar
  46. 46.
    Dillencourt, M.B.: Realizability of Delaunay triangulations. Inform. Process. Lett. 33(6), 283–287 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  47. 47.
    Dillencourt, M.B., Eppstein, D., Hirschberg, D.S.: Geometric thickness of complete graphs. J. Graph Algorithms & Applications 4(3), 5–17 (2000)zbMATHMathSciNetGoogle Scholar
  48. 48.
    Dillencourt, M.B., Smith, W.D.: Graph-theoretical conditions for inscribability and Delaunay realizability. In: Proc. 6th Canad. Conf. Comput. Geom., pp. 287–292 (1994)Google Scholar
  49. 49.
    Dujmović, V., Fellows, M., Hallett, M., Kitching, M., Liotta, G., McCartin, C., Nishimura, N., Radge, P., Rosamond, F., Suderman, M., Whitesides, S., Wood, D.: On the parameterized complexity of layered graph drawing. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 488–499. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  50. 50.
    Dujmović, V., Morin, P., Wood, D.: Pathwidth and three-dimensional straight line grid drawings of graphs. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 42–53. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  51. 51.
    Dujmović, V., Wood, D.R.: Tree-partitions of k-trees with applications in graph layout. Technical Report 02-03, Dept. Comput. Sci., Carleton Univ., Ottawa, Canada (2002)Google Scholar
  52. 52.
    Dujmović, V., Wood, D.R.: On linear layouts of graphs. Technical Report 03-05, Dept. Comput. Sci., Carleton Univ., Ottawa, Canada (2003)Google Scholar
  53. 53.
    Dujmović, V., Wood, D.R.: Stacks, queues and tracks: Layouts of graphs subdivisions. Technical Report 03-07, Dept. Comput. Sci., Carleton Univ., Ottawa, Canada (2003)Google Scholar
  54. 54.
    Dujmović, V., Wood, D.R.: Three-dimensional grid drawings with subquadratic volume. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 190–201. Springer, Heidelberg (2004); These processingCrossRefGoogle Scholar
  55. 55.
    Dujmović, V., Wood, D.R.: Track layouts of graphs. Technical Report 03-06, Dept. Comput. Sci., Carleton Univ., Ottawa, Canada (2003)Google Scholar
  56. 56.
    Dujmović, V., Wood, D.R.: Tree-partitions of k-trees with application in graph layout. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 205–217. Springer, Heidelberg (2003) (to appear)CrossRefGoogle Scholar
  57. 57.
    Eades, P., Whitesides, S.: The realization problem for Euclidean minimum spanning trees is NP-hard. Algorithmica 16, 60–82 (1996); (Di Battista, G., Tamassia, R.(ed.) special issue on Graph Drawing)zbMATHMathSciNetCrossRefGoogle Scholar
  58. 58.
    Edler, B.: Effiziente Algorithmen für flächenminimale Layouts von Bäumen. Diplomarbeit, Universität Passau (1999)Google Scholar
  59. 59.
    ElGindy, H., Avis, D., Toussaint, G.T.: Applications of a two-dimensional hidden-line algorithm to other geometric problems. Computing 31, 191–202 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  60. 60.
    Even, G., Guha, S., Schieber, B.: Improved approximations of crossings in graph drawing and VLSI layout area. In: Proceedings of the 32nd ACM Symposium on Theory of Computing (STOC 2000), pp. 296–305. ACM Press, New York (2000)CrossRefGoogle Scholar
  61. 61.
    Everett, H.: Visibility graph recognition. Report 231/90, Dept. Comput. Sci., Univ. Toronto, Toronto, ON. Ph.D. Thesis (1990)Google Scholar
  62. 62.
    Felsner, S.: Convex drawings of planar graphs and the order dimension of 3-polytopes. Order 18(1), 19–37 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    Felsner, S., Liotta, G., Wismath, S.: Straight line drawings on restricted integer grids in two and three dimensions. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 328–342. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  64. 64.
    Feng, Q.-W., Cohen, R.F., Eades, P.: Planarity for clustered graphs. In: Spirakis, P.G. (ed.) ESA 1995. LNCS, vol. 979, pp. 213–226. Springer, Heidelberg (1995)Google Scholar
  65. 65.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)zbMATHGoogle Scholar
  66. 66.
    Garey, M.R., Johnson, D.S., Preparata, F.P., Tarjan, R.E.: Triangulating a simple polygon. Inform. Process. Lett. 7, 175–179 (1978)zbMATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    Garg, A.: New results on drawing angle graphs. Comput. Geom. Theory Appl. 9(1-2), 43–82 (1998)zbMATHMathSciNetGoogle Scholar
  68. 68.
    Garg, A., Goodrich, M.T., Tamassia, R.: Planar upward tree drawings with optimal area. Internat. J. Comput. Geom. Appl. 6, 333–356 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  69. 69.
    Garg, A., Liotta, G.: Almost bend-optimal planar orthogonal drawings of biconnected degree-3 planar graphs in quadratic time. In: Kratochvíl, J. (ed.) GD 1999. LNCS, vol. 1731, p. 38. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  70. 70.
    Garg, A., Rusu, A.: Straight-line drawings of binary trees with linear area and good aspect ratio. In: Goodrich, M. (ed.) GD 2002. LNCS, vol. 2528, pp. 320–331. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  71. 71.
    Garg, A., Rusu, A.: Area-efficient drawings of outerplanar graphs. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 129–134. Springer, Heidelberg (2004); These proceedingsCrossRefGoogle Scholar
  72. 72.
    Garg, A., Tamassia, R.: A new minimum cost flow algorithm with applications to graph drawing. In: North, S.C. (ed.) GD 1996. LNCS, vol. 1190, Springer, Heidelberg (1997)Google Scholar
  73. 73.
    Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31(2), 601–625 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  74. 74.
    Garg, A., Tamassia, R., Vocca, P.: Drawing with colors. In: Díaz, J. (ed.) ESA 1996. LNCS, vol. 1136, pp. 12–26. Springer, Heidelberg (1996)Google Scholar
  75. 75.
    Ghosh, S.K.: On recognizing and characterizing visibility graphs of simple polygons. Discrete Comput. Geom. 17, 143–162 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  76. 76.
    Gregori, A.: Unit length embedding of binary trees on a square grid. Inform. Process. Lett. 31, 167–172 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  77. 77.
    Grohe, M.: Computing crossing numbers in quadratic time. In: Proceedings of the 32nd ACM Symposium on Theory of Computing (STOC 2000), pp. 231–236. ACM Press, New York (2000)Google Scholar
  78. 78.
    Gutwenger, C., Jünger, M., Leipert, S., Mutzel, P., Percan, M., Weiskircher, R.: Advances in c-planarity testing of clustered graphs. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 220–325. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  79. 79.
    Gutwenger, C., Mutzel, P.: An experimental study of crossing minimization heuristics. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 13–24. Springer, Heidelberg (2004); These proceedingsCrossRefGoogle Scholar
  80. 80.
    Gutwenger, C., Mutzel, P.: Graph embedding with minimum depth and maximum external face. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 259–272. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  81. 81.
    Gutwenger, C., Mutzel, P., Weiskircher, R.: Inserting an edge into a planar graph. In: Proc. Ninth Annual ACM-SIAM Symp. Discrete Algorithms (SODA 2001), Washington, DC, pp. 246–255. ACM Press, New York (2001)Google Scholar
  82. 82.
    Healy, P., Kuusik, A.: The vertex-exchange graph: a new concept for multi-level crossing minimization. In: Kratochvíl, J. (ed.) GD 1999. LNCS, vol. 1731, pp. 205–216. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  83. 83.
    Hertel, S., Mehlhorn, K.: Fast triangulation of the plane with respect to simple polygons. Inform. Control 64, 52–76 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  84. 84.
    Jaromczyk, J.W., Toussaint, G.T.: Relative neighborhood graphs and their relatives. Proc. IEEE 80(9), 1502–1517 (1992)CrossRefGoogle Scholar
  85. 85.
    Jünger, M., Lee, E., Mutzel, P., Odenthal, T.: A polyhedral approach to the multi-layer crossing number problem. In: DiBattista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 13–24. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  86. 86.
    Kant, G.: A new method for planar graph drawings on a grid. In: Proc. 33rd Annu. IEEE Sympos. Found. Comput. Sci., pp. 101–110 (1992)Google Scholar
  87. 87.
    Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16, 4–32 (1996); (Di Battista, G., Tamassia, R. (ed.) special issue on Graph Drawing)zbMATHMathSciNetCrossRefGoogle Scholar
  88. 88.
    Kaufmann, M., Wiese, R.: Embedding vertices at points: Few bends suffice for planar graphs. Journal of Graph Algorithms and Applications 6(1), 115–129 (2002)zbMATHMathSciNetGoogle Scholar
  89. 89.
    Keil, M.: Computing a subgraph of the minimum weight triangulation. Comput. Geom. Theory Appl. 4, 13–26 (1994)zbMATHMathSciNetGoogle Scholar
  90. 90.
    Kirkpatrick, D.G.: A note on Delaunay and optimal triangulations. Inform. Process. Lett. 10(3), 127–128 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  91. 91.
    Klau, G.W., Klein, K., Mutzel, P.: An experimental comparison of orthogonal compaction algorithms. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 37–51. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  92. 92.
    Klau, G.W., Mutzel, P.: Optimal compaction of orthogonal grid drawings. In: Cornuéjols, G., Burkard, R.E., Woeginger, G.J. (eds.) IPCO 1999. LNCS, vol. 1610, pp. 304–319. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  93. 93.
    Leighton, F.T.: New lower bound techniques for VLSI. In: Proc. 22nd Annu. IEEE Sympos. Found. Comput. Sci., pp. 1–12 (1981)Google Scholar
  94. 94.
    Leighton, F.T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM 46(6), 787–832 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  95. 95.
    Lenhart, W., Liotta, G.: Drawing outerplanar minimum weight triangulations. Inform. Process. Lett. 57(5), 253–260 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  96. 96.
    Lenhart, W., Liotta, G.: Proximity drawings of outerplanar graphs. In: North, S.C. (ed.) GD 1996. LNCS, vol. 1190, Springer, Heidelberg (1997)Google Scholar
  97. 97.
    Lenhart, W., Liotta, G.: The drawability problem for minimum weight triangulations. Theoret. Comp. Sci. 270, 261–286 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  98. 98.
    Levcopoulos, C., Krznaric, D.: Tight lower bounds for minimum weight triangulation heuristic. Inform. Process. Lett. 57, 129–135 (1996)MathSciNetCrossRefGoogle Scholar
  99. 99.
    Levcopoulos, C., Krznaric, D.: A linear-time approximation schema for minimum weight triangulation of convex polygons. Algorithmica 21, 285–311 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  100. 100.
    Lingas, A.: A new heuristic for minimum weight triangulation. SIAM J. Algebraic Discrete Methods 8(4), 646–658 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  101. 101.
    Liotta, G., Di Battista, G.: Computing proximity drawings of trees in the 3-dimensional space. In: Sack, J.-R., Akl, S.G., Dehne, F., Santoro, N. (eds.) WADS 1995. LNCS, vol. 955, pp. 239–250. Springer, Heidelberg (1995)Google Scholar
  102. 102.
    Liotta, G., Lubiw, A., Meijer, H., Whitesides, S.H.: The rectangle of influence drawability problem. Comput. Geom. Theory Appl. 10(1), 1–22 (1998)zbMATHMathSciNetGoogle Scholar
  103. 103.
    Liotta, G., Meijer, H.: Voronoi drawings of trees. Comput. Geom. Theory Appl. (2003) (to appear)Google Scholar
  104. 104.
    Lovasz, L., Vesztergombi, K., Wagner, U., Welzl, E.: Convex quadrilaterals and k-sets. Contemporary Mathematics (2003)Google Scholar
  105. 105.
    Lubiw, A., Sleumer, N.: Maximal outerplanar graphs are relative neighborhood graphs. In: Proc. 5th Canad. Conf. Comput. Geom., pp. 198–203 (1993)Google Scholar
  106. 106.
    Malitz, S., Papakostas, A.: On the angular resolution of planar graphs. In: Proc. 24th Annu. ACM Sympos. Theory Comput., pp. 527–538 (1992)Google Scholar
  107. 107.
    Manacher, G.K., Zobrist, A.L.: Neither the greedy nor the Delaunay triangulation of a planar point set approximates the optimal triangulation. Inform. Process. Lett. 9, 31–34 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  108. 108.
    Mansfield, A.: Determining the thickness of a graph is NP-hard. Math. Proc. Cambridge Philos. Soc. 93(9), 9–23 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  109. 109.
    Mehlhorn, K., Näher, T.S.S., Schirra, S., Seel, M., Seidel, R., Uhrig, C.: Checking geometric programs or verification of geometric structures. Comput. Geom. Theory Appl. 12, 85–113 (1999)zbMATHGoogle Scholar
  110. 110.
    Monma, C., Suri, S.: Transitions in geometric minimum spanning trees. Discrete Comput. Geom. 8, 265–293 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  111. 111.
    Mutzel, P., Weiskircher, R.: Computing optimal embeddings for planar graphs. In: Du, D.-Z., Eades, P., Sharma, A.K., Lin, X., Estivill-Castro, V. (eds.) COCOON 2000. LNCS, vol. 1858, pp. 95–104. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  112. 112.
    Pach, J., Shahrokhi, F., Szegedy, M.: Applications of the crossing number. Algorithmica 16, 111–117 (1996) (Di Battista, G., Tamassia, R.(ed.) special issue on Graph Drawing)zbMATHMathSciNetCrossRefGoogle Scholar
  113. 113.
    Pach, J., Thiele, T., Tóth, G.: Three-dimensional grid drawings of graphs. In: DiBattista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 47–51. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  114. 114.
    Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17, 427–439 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  115. 115.
    Pizzonia, M., Tamassia, R.: Minimum depth graph embedding. In: Paterson, M. (ed.) ESA 2000. LNCS, vol. 1879, pp. 356–367. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  116. 116.
    Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction, 3rd edn. Springer, Heidelberg (October 1990)Google Scholar
  117. 117.
    Rahman, S., Egi, N., Nishizeki, T.: No-bend orthogonal drawings of subdivisions of planar triconnected cubic graphs. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 387–392. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  118. 118.
    Rappaport, D., Meijer, H.: Computing the minimum weight triangulation of a set of linearly ordered points. Information Processing Letters 42, 35–38 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  119. 119.
    Richter, R., Thomassen, C.: Relations between crossing numbers of complete and complete bipartite graphs (February 1997)Google Scholar
  120. 120.
    Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete Comput. Geom. 1(4), 343–353 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  121. 121.
    Scheinerman, E.R., Wilf, H.S.: The rectilinear crossing number of a complete graph and Sylvester’s “four point problem” of geometric probability. Amer. Math. Monthly 101(10), 939–943 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  122. 122.
    Schnyder, W.: Planar graphs and poset dimension. Order 5, 323–343 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  123. 123.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proc. 1st ACM-SIAM Sympos. Discrete Algorithms, pp. 138–148 (1990)Google Scholar
  124. 124.
    Schnyder, W., Trotter, W.T.: Convex embeddings of 3-connected plane graphs. Abstracts of the AMS 13(5), 502 (1992)Google Scholar
  125. 125.
    Shin, C.-S., Kim, S.K., Chwa, K.-Y.: Area-efficient algorithms for straight-line tree drawings. Comput. Geom. Theory Appl. 15, 175–202 (2000)zbMATHMathSciNetGoogle Scholar
  126. 126.
    Sugiyama, K., Tagawa, S., Toda, M.: Methods for visual understanding of hierarchical systems. IEEE Trans. Syst. Man Cybern. SMC-11(2), 109–125 (1981)MathSciNetCrossRefGoogle Scholar
  127. 127.
    Sýkora, O., Vrťo, I.: On VLSI layouts of the star graph and related networks. The VLSI Journal 17, 83–93 (1994)zbMATHCrossRefGoogle Scholar
  128. 128.
    Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3), 421–444 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  129. 129.
    Toussaint, G.: Efficient triangulation of simple polygons. Visual Comput. 7, 280–295 (1991)CrossRefGoogle Scholar
  130. 130.
    Vijayan, G., Wigderson, A.: Rectilinear graphs and their embeddings. SIAM J. Comput. 14, 355–372 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  131. 131.
    Vijayan, V.: Geometry of planar graphs with angles. In: Proc. 2nd Annu. ACM Sympos. Comput. Geom., pp. 116–124 (1986)Google Scholar
  132. 132.
    Waddle, V., Malhotra, A.: An E log E line crossing algorithm for levelled graphs. In: Kratochvíl, J. (ed.) GD 1999. LNCS, vol. 1731, pp. 59–70. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  133. 133.
    Wang, C.A., Chin, F.Y., Yang, B.: Triangulations without minimum weight drawing. In: Bongiovanni, G., Petreschi, R., Gambosi, G. (eds.) CIAC 2000. LNCS, vol. 1767, pp. 163–173. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  134. 134.
    Woo, T.C., Shin, S.Y.: A linear time algorithm for triangulating a point-visible polygon. ACM Trans. Graph. 4(1), 60–70 (1985)zbMATHGoogle Scholar
  135. 135.
    Wood, D.R.: Queue layouts, tree-width, and three-dimensional graph drawing. In: Agrawal, M., Seth, A.K. (eds.) FSTTCS 2002. LNCS, vol. 2556, pp. 348–359. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  136. 136.
    Zhang, H., He, X.: Compact visibility representations and straight-line grid embedding of plane graphs. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 493–504. Springer, Heidelberg (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Franz Brandenburg
    • 1
  • David Eppstein
    • 1
  • Michael T. Goodrich
    • 2
  • Stephen Kobourov
    • 3
  • Giuseppe Liotta
    • 4
  • Petra Mutzel
    • 5
  1. 1.Univ. of PassauGermany
  2. 2.Univ. of California-IrvineUSA
  3. 3.Univ. of ArizonaUSA
  4. 4.Univ. of PerugiaItaly
  5. 5.Vienna Univ. of TechnologyAustria

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