Queue Layouts, Tree-Width, and Three-Dimensional Graph Drawing

  • David R. Wood
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2556)


A three-dimensional (straight-line grid) drawing of a graph represents the vertices by points in Z3 and the edges by non-crossing line segments. This research is motivated by the following open problem due to Felsner, Liotta, and Wismath [Graph Drawing ’01, Lecture Notes in Comput. Sci., 2002]: does every n-vertex planar graph have a threedimensional drawing with O(n) volume? We prove that this question is almost equivalent to an existing one-dimensional graph layout problem. A queue layout consists of a linear order σ of the vertices of a graph, and a partition of the edges into queues, such that no two edges in the same queue are nested with respect to σ. The minimum number of queues in a queue layout of a graph is its queue-number. Let G be an n-vertex member of a proper minor-closed family of graphs (such as a planar graph). We prove that G has a O(1) × O(1) × O(n) drawing if and only if G has O(1) queue-number. Thus the above question is almost equivalent to an open problem of Heath, Leighton, and Rosenberg [SIAM J. Discrete Math., 1992], who ask whether every planar graph has O(1) queue-number? We also present partial solutions to an open problem of Ganley and Heath [Discrete Appl. Math., 2001], who ask whether graphs of bounded tree-width have bounded queue-number? We prove that graphs with bounded path-width, or both bounded tree-width and bounded maximum degree, have bounded queue-number. As a corollary we obtain three-dimensional drawings with optimal O(n) volume, for series-parallel graphs, and graphs with both bounded tree-width and bounded maximum degree.


Planar Graph Outerplanar Graph Single Queue Grid Drawing Book Embedding 
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  1. [1]
    N. Alon, C. McDiarmid, and B. Reed, Acyclic coloring of graphs. Random Structures Algorithms, 2(3):277–288, 1991.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    H. L. Bodlaender, A partial k-arboretum of graphs with bounded treewidth. Theoret. Comput. Sci., 209(1–2):1–45, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    H. L. Bodlaender and J. Engelfriet, Domino treewidth. J. Algorithms, 24(1):94–123, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    T. Calamoneri and A. Sterbini, 3D straight-line grid drawing of 4-colorable graphs. Inform. Process. Lett., 63(2):97–102, 1997.CrossRefMathSciNetGoogle Scholar
  5. [5]
    R. F. Cohen, P. Eades, T. Lin, and F. Ruskey, Three-dimensional graph drawing. Algorithmica, 17(2):199–208, 1996.CrossRefMathSciNetGoogle Scholar
  6. [6]
    H. de Fraysseix, J. Pach, and R. Pollack, How to draw a planar graph on a grid. Combinatorica, 10(1):41–51, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    E. di Giacomo, G. Liotta, and S. Wismath, Drawing series-parallel graphs on a box. In S. Wismath, ed., Proc. 14th Canadian Conf. on Computational Geometry (CCCG’ 02), The University of Lethbridge, Canada, 2002.Google Scholar
  8. [8]
    J. Díaz, J. Petit, and M. Serna, A survey of graph layout problems. ACM Comput. Surveys, to appear.Google Scholar
  9. [9]
    R. P. Dilworth, A decomposition theorem for partially ordered sets. Ann. of Math. (2), 51:161–166, 1950.CrossRefMathSciNetGoogle Scholar
  10. [10]
    G. Ding and B. Oporowski, Some results on tree decomposition of graphs. J. Graph Theory, 20(4):481–499, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    G. Ding and B. Oporowski, On tree-partitions of graphs. Discrete Math., 149(1–3):45–58, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    V. Dujmović, P. Morin, and D. R. Wood, Path-width and three-dimensional straight-line grid drawings of graphs. In M. Goodrich, ed., Proc. 10th International Symp. on Graph Drawing (GD’ 02), Lecture Notes in Comput. Sci., Springer, to appear.Google Scholar
  13. [13]
    V. Dujmović and D. R. Wood, Tree-partitions of k-trees with applications in graph layout. Tech. Rep. TR-02-03, School of Computer Science, Carleton University, Ottawa, Canada, 2002.Google Scholar
  14. [14]
    S. Felsner, S. Wismath, and G. Liotta, Straight-line drawings on restricted integer grids in two and three dimensions. In P. Mutzel, M. Jünger, and S. Leipert, eds., Proc. 9th International Symp. on Graph Drawing (GD’ 01), vol. 2265 of Lecture Notes in Comput. Sci., pp. 328–342, Springer, 2002.Google Scholar
  15. [15]
    G. Fertin, A. Raspaud, and B. Reed, On star coloring of graphs. In A. Branstädt and V. B. Le, eds., Proc. 27th International Workshop on Graph-Theoretic Concepts in Computer Science (WG’ 01), vol. 2204 of Lecture Notes in Comput. Sci., pp. 140–153, Springer, 2001.Google Scholar
  16. [16]
    J. L. Ganley and L. S. Heath, The pagenumber of k-trees is O(k). Discrete Appl. Math., 109(3):215–221, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    M. R. Garey, D. S. Johnson, G. L. Miller, and C. H. Papadimitriou, The complexity of coloring circular arcs and chords. SIAM J. Algebraic Discrete Methods, 1(2):216–227, 1980.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    R. Halin, Tree-partitions of infinite graphs. Discrete Math., 97:203–217, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    L. S. Heath, F. T. Leighton, and A. L. Rosenberg, Comparing queues and stacks as mechanisms for laying out graphs. SIAM J. Discrete Math., 5(3):398–412, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    L. S. Heath and A. L. Rosenberg, Laying out graphs using queues. SIAM J. Comput., 21(5):927–958, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    S. M. Malitz, Graphs with E edges have pagenumber O(√E). J. Algorithms, 17(1):71–84, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    J. Nesetril and P. Ossona de Mendez, Colorings and homomorphisms of minor closed classes. Tech. Rep. 2001-025, Institut Teoretické Informatiky, Universita Karlova v Praze, Czech Republic, 2001.Google Scholar
  23. [23]
    J. Pach, T. Thiele, and G. Tóth, Three-dimensional grid drawings of graphs. In G. Di Battista, ed., Proc. 5th International Symp. on Graph Drawing (GD’ 97), vol. 1353 of Lecture Notes in Comput. Sci., pp. 47–51, Springer, 1998.Google Scholar
  24. [24]
    S. V. Pemmaraju, Exploring the Powers of Stacks and Queues via Graph Layouts. Ph.D. thesis, Virginia Polytechnic Institute and State University, Virginia, U.S.A., 1992.Google Scholar
  25. [25]
    T. Poranen, A new algorithm for drawing series-parallel digraphs in 3D. Tech. Rep. A-2000-16, Dept. of Computer and Information Sciences, University of Tampere, Finland, 2000.Google Scholar
  26. [26]
    S. Rengarajan and C. E. Veni Madhavan, Stack and queue number of 2-trees. In D. Ding-Zhu and L. Ming, eds., Proc. 1st Annual International Conf. on Computing and Combinatorics (COCOON’ 95), vol. 959 of Lecture Notes in Comput. Sci., pp. 203–212, Springer, 1995.Google Scholar
  27. [27]
    W. Schnyder, Planar graphs and poset dimension. Order, 5(4):323–343, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    D. Seese, Tree-partite graphs and the complexity of algorithms. In L. Budach, ed., Proc. International Conf. on Fundamentals of Computation Theory, vol. 199 of Lecture Notes in Comput. Sci., pp. 412–421, Springer, 1985.Google Scholar
  29. [29]
    F. Shahrokhi and W. Shi, On crossing sets, disjoint sets, and pagenumber. J. Algorithms, 34(1):40–53, 2000.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • David R. Wood
    • 1
  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada

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