Three-Dimensional Grid Drawings with Sub-quadratic Volume

  • Vida Dujmović
  • David R. Wood
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2912)

Abstract

A three-dimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line-segments representing the edges are pairwise non-crossing. A \(\mathcal{O}(n^{3/2})\) volume bound is proved for three-dimensional grid drawings of graphs with bounded degree, graphs with bounded genus, and graphs with no bounded complete graph as a minor. The previous best bound for these graph families was \(\mathcal{O}(n^{2})\). These results (partially) solve open problems due to Pach, Thiele, and Tóth [Graph Drawing 1997] and Felsner, Liotta, and Wismath [Graph Drawing 2001].

References

  1. 1.
    Alon, N., Seymour, P., Thomas, R.: A separator theorem for nonplanar graphs. J. Amer. Math. Soc. 3(4), 801–808 (1990)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theoret. Comput. Sci. 209(1-2), 1–45 (1998)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bose, P., Czyzowicz, J., Morin, P., Wood, D.R.: The maximum number of edges in a three-dimensional grid-drawing. In: Proc. 19th European Workshop on Computational Geometry, Germany, pp. 101–103. Univ. of Bonn. (2003)Google Scholar
  4. 4.
    Calamoneri, T., Sterbini, A.: 3D straight-line grid drawing of 4-colorable graphs. Inform. Process. Lett. 63(2), 97–102 (1997)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Cohen, R.F., Eades, P., Lin, T., Ruskey, F.: Threedimensional graph drawing. Algorithmica 17(2), 199–208 (1996)CrossRefMathSciNetGoogle Scholar
  6. 6.
    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
  7. 7.
    Giacomo, E.D., Liotta, G., Wismath, S.: Drawing seriesparallel graphs on a box. In: Proc. 14th Canadian Conf. on Computational Geometry (CCCG 2002), pp. 149–153. The Univ. of Lethbridge, Canada (2002)Google Scholar
  8. 8.
    Giacomo, E.D., 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
  9. 9.
    Djidjev, H.N.: A separator theorem. C. R. Acad. Bulgare Sci. 34(5), 643–645 (1981)MATHMathSciNetGoogle Scholar
  10. 10.
    Dujmović, V., Morin, P., Wood, D.R.: Path-width and threedimensional 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
  11. 11.
    Dujmović, V., Wood, D.R.: New results in graph layout. Tech. Report TR-2003-04, School of Computer Science, Carleton Univ., Ottawa, Canada (2003)Google Scholar
  12. 12.
    Dujmović, V., Wood, D.R.: Tree-partitions of k-trees with applications in graph layout. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 205–217. Springer, Heidelberg (2003) (to appear)CrossRefGoogle Scholar
  13. 13.
    Erdős, P., Lovász, L.: Problems and results on 3-chromatic hypergraphs and some related questions. In: Infinite and Finite Sets. Colloq. Math. Soc. János Bolyai, vol. 10, pp. 609–627. North-Holland, Amsterdam (1975)Google Scholar
  14. 14.
    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
  15. 15.
    Fertin, G., Raspaud, A., Reed, B.: On star coloring of graphs. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 140–153. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  16. 16.
    Gilbert, J.R., Hutchinson, J.P., Tarjan, R.E.: A separator theorem for graphs of bounded genus. J. Algorithms 5(3), 391–407 (1984)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gyárfás, A., West, D.: Multitrack interval graphs. In: Proc. 26th Southeastern Int’l Conf. on Combinatorics, Graph Theory and Computing. Congr. Numer., vol. 109, pp. 109–116 (1995)Google Scholar
  18. 18.
    Hasunuma, T.: Laying out iterated line digraphs using queues. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 202–213. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  19. 19.
    Heath, L.S., Leighton, F.T., Rosenberg, A.L.: Comparing queues and stacks as mechanisms for laying out graphs. SIAM J. Discrete Math. 5(3), 398–412 (1992)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Heawood, P.J.: Map colour theorem. Quart. J. Pure Appl. Math. 24, 332–338 (1890)Google Scholar
  21. 21.
    Kostochka, A.V.: The minimum Hadwiger number for graphs with a given mean degree of vertices. Metody Diskret. Analiz. 38, 37–58 (1982)MATHMathSciNetGoogle Scholar
  22. 22.
    Molloy, M., Reed, B.: Graph colouring and the probabilistic method. Algorithms and Combinatorics, vol. 23. Springer, Heidelberg (2002)MATHGoogle Scholar
  23. 23.
    Nešetřil, J., de Mendez, P.O.: Colorings and homomorphisms of minor closed classes. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry, The Goodman-Pollack Festschrift. Algorithms and Combinatorics, vol. 25, Springer, Heidelberg (2003)Google Scholar
  24. 24.
    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); Also in Chazelle, B., Goodman, J.E., Pollack, R. (eds.): Advances in discrete and computational geometry. Contempory Mathematics, vol. 223, pp. 251–255. Amer. Math. Soc., Providence (1999)CrossRefGoogle Scholar
  25. 25.
    Poranen, T.: A new algorithm for drawing series-parallel digraphs in 3D. Tech. Report A-2000-16, Dept. of Computer and Information Sciences, Univ. of Tampere, Finland (2000)Google Scholar
  26. 26.
    Thomason, A.: An extremal function for contractions of graphs. Math. Proc. Cambridge Philos. Soc. 95(2), 261–265 (1984)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    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

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Vida Dujmović
    • 1
  • David R. Wood
    • 2
  1. 1.School of Computer ScienceMcGill UniversityMontréalCanada
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada

Personalised recommendations