Bounded Degree Book Embeddings and Three-Dimensional Orthogonal Graph Drawing

  • David R. Wood
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2265)

Abstract

A book embedding of a graph consists of a linear ordering of the vertices along a line in 3-space (the spine), and an assignment of edges to half-planes with the spine as boundary (the pages), so that edges assigned to the same page can be drawn on that page without crossings. Given a graph G = (V,E), let f : V → ℕ be a function such that 1 ≤ f(υ) ≤ deg(υ). We present a Las Vegas algorithm which produces a book embedding of G with \( O(\sqrt {|E| \cdot \max _\upsilon \left\lceil {\deg (\upsilon )/f(\upsilon )} \right\rceil } ) \) pages, such that at most f(v) edges incident to a vertex v are on a single page. This algorithm generalises existing results for book embeddings. We apply this algorithm to produce 3-D orthogonal drawings with one bend per edge and O(∣V3/2E∣) volume, and single-row drawings with two bends per edge and the same volume. In the produced drawings each edge is entirely contained in some Z-plane; such drawings are without so-called cross-cuts, and are particularly appropriate for applications in multilayer VLSI. Using a different approach, we achieve two bends per edge with O(∣V ∣∣E∣) volume but with cross-cuts. These results establish improved bounds for the volume of 3-D orthogonal graph drawings.

References

  1. 2.
    A. Aggarwal, M. Klawe, and P. Shor. Multilayer grid embeddings for VLSI. Algorithmica, 6(1):129–151, 1991.MATHCrossRefMathSciNetGoogle Scholar
  2. 3.
    T. Biedl. 1-bend 3-D orthogonal box-drawings: Two open problems solved. J. Graph Algorithms Appl., 5(3):1–15, 2001.MathSciNetGoogle Scholar
  3. 4.
    T. Biedl, T. Thiele, and D. R. Wood. Three-dimensional orthogonal graph drawing with optimal volume. In J. Marks, editor, Proc. Graph Drawing: 8th International Symp. (GD’00), volume 1984 of Lecture Notes in Comput. Sci., pages 284–295. Springer, 2001.Google Scholar
  4. 5.
    T. C. Biedl. Three approaches to 3D-orthogonal box-drawings. In S. Whitesides, editor, Proc. Graph Drawing: 6th International Symp. (GD’98), volume 1547 of Lecture Notes in Comput. Sci., pages 30–43. Springer, 1998.Google Scholar
  5. 6.
    T. C. Biedl and M. Kaufmann. Area-efficient static and incremental graph drawings. In R. Burkhard and G. Woeginger, editors, Proc. Algorithms: 5th Annual European Symp. (ESA’97), volume 1284 of Lecture Notes in Comput. Sci., pages 37–52. Springer, 1997.Google Scholar
  6. 7.
    T. C. Biedl, T. Shermer, S. Whitesides, and S. Wismath. Bounds for orthogonal 3-D graph drawing. J. Graph Algorithms Appl., 3(4):63–79, 1999.MATHMathSciNetGoogle Scholar
  7. 8.
    F. R. K. Chung, F. T. Leighton, and A. L. Rosenberg. Embedding graphs in books: a layout problem with applications to VLSI design. SIAM J. Algebraic Discrete Methods, 8(1):33–58, 1987.MATHCrossRefMathSciNetGoogle Scholar
  8. 9.
    R. P. Dilworth. A decomposition theorem for partially ordered sets. Ann. of Math. (2), 51:161–166, 1950.CrossRefMathSciNetGoogle Scholar
  9. 10.
    S. L. Hakimi and O. Kariv. A generalization of edge-coloring in graphs. J. Graph Theory, 10(2):139–154, 1986.MATHCrossRefMathSciNetGoogle Scholar
  10. 11.
    L. S. Heath and A. L. Rosenberg. Laying out graphs using queues. SIAM J. Comput., 21(5):927–958, 1992.MATHCrossRefMathSciNetGoogle Scholar
  11. 12.
    E. S. Kuh, T. Kashiwabara, and T. Fujisawa. On optimum single-row routing. IEEE Trans. Circuits and Systems, 26(6):361–368, 1979.MATHCrossRefMathSciNetGoogle Scholar
  12. 13.
    S. M. Malitz. Genus g graphs have pagenumber O(√g). J. Algorithms, 17(1):85–109, 1994.MATHCrossRefMathSciNetGoogle Scholar
  13. 14.
    S. M. Malitz. Graphs with E edges have pagenumber O(√E). J. Algorithms, 17(1):71–84, 1994.MATHCrossRefMathSciNetGoogle Scholar
  14. 15.
    F. Malucelli and S. Nicoloso. Optimal partition of a bipartite graph into noncrossing matchings. In [1].Google Scholar
  15. 16.
    R. Tarjan. Sorting using networks of queues and stacks. J. Assoc. Comput. Mach., 19:341–346, 1972.MATHMathSciNetGoogle Scholar
  16. 17.
    V. G. Vizing. On an estimate of the chromatic class of a p-graph. Diskret. Analiz No., 3:25–30, 1964.MathSciNetGoogle Scholar
  17. 18.
    D. R. Wood. Three-dimensional orthogonal graph drawings with one bend per edge. Submitted. See Three-Dimensional Orthogonal Graph Drawing. Ph.D. thesis, School of Computer Science and Software Engineering, Monash University, Australia, 2000.Google Scholar
  18. 19.
    D. R. Wood. Geometric thickness in a grid of linear area. In [1].Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • David R. Wood
    • 1
  1. 1.Basser Department of Computer ScienceThe University of SydneySydneyAustralia

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