# Bounded Degree Book Embeddings and Three-Dimensional Orthogonal Graph Drawing

## 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*(∣*V* ∣^{3/2}∣*E*∣) 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.

## Keywords

Bipartite Graph Edge Incident Edge Route Single Page Canonical Ordering## References

- 2.A. Aggarwal, M. Klawe, and P. Shor. Multilayer grid embeddings for VLSI.
*Algorithmica*, 6(1):129–151, 1991.zbMATHCrossRefMathSciNetGoogle Scholar - 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 - 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 - 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 - 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 - 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.zbMATHMathSciNetGoogle Scholar - 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.zbMATHCrossRefMathSciNetGoogle Scholar - 9.R. P. Dilworth. A decomposition theorem for partially ordered sets.
*Ann. of Math. (2)*, 51:161–166, 1950.CrossRefMathSciNetGoogle Scholar - 10.S. L. Hakimi and O. Kariv. A generalization of edge-coloring in graphs.
*J. Graph Theory*, 10(2):139–154, 1986.zbMATHCrossRefMathSciNetGoogle Scholar - 11.L. S. Heath and A. L. Rosenberg. Laying out graphs using queues.
*SIAM J. Comput.*, 21(5):927–958, 1992.zbMATHCrossRefMathSciNetGoogle Scholar - 12.E. S. Kuh, T. Kashiwabara, and T. Fujisawa. On optimum single-row routing.
*IEEE Trans. Circuits and Systems*, 26(6):361–368, 1979.zbMATHCrossRefMathSciNetGoogle Scholar - 13.S. M. Malitz. Genus
*g*graphs have pagenumber*O(√g)*.*J. Algorithms*, 17(1):85–109, 1994.zbMATHCrossRefMathSciNetGoogle Scholar - 14.S. M. Malitz. Graphs with
*E*edges have pagenumber*O(√E)*.*J. Algorithms*, 17(1):71–84, 1994.zbMATHCrossRefMathSciNetGoogle Scholar - 15.F. Malucelli and S. Nicoloso. Optimal partition of a bipartite graph into noncrossing matchings. In [1].Google Scholar
- 16.R. Tarjan. Sorting using networks of queues and stacks.
*J. Assoc. Comput. Mach.*, 19:341–346, 1972.zbMATHMathSciNetGoogle Scholar - 17.V. G. Vizing. On an estimate of the chromatic class of a
*p*-graph.*Diskret. Analiz No.*, 3:25–30, 1964.MathSciNetGoogle Scholar - 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 - 19.D. R. Wood. Geometric thickness in a grid of linear area. In [1].Google Scholar