GD 1999: Graph Drawing pp 311-322

# Multi-dimensional Orthogonal Graph Drawing with Small Boxes

Extended Abstract
• David R. Wood
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1731)

## Abstract

In this paper we investigate the general position model for the drawing of arbitrary degree graphs in the D-dimensional (D ≥ 2) orthogonal grid. In this model no two vertices lie in the same grid hyperplane. We show that for D ≥ 3, given an arbitrary layout and initial edge routing a crossing-free orthogonal drawing can be determined. We distinguish two types of algorithms. Our layout-based algorithm, given an arbitrary fixed layout, determines a degree-restricted orthogonal drawing with each vertex having aspect ratio two. Using a balanced lay-out this algorithm establishes improved bounds on the size of vertices for 2-D and 3-D drawings. Our routing-based algorithm produces 2-degree- restricted 3-D orthogonal drawings. One advantage of our approach in 3-D is that edges are typically routed on each face of a vertex; hence the produced drawings are more truly three-dimensional than those produced by some existing algorithms.

## Keywords

Graph Drawing Edge Route Orthogonal Grid Orthogonal Graph Positive Vertex
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.

## References

1. [1]
T. 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, Berlin, 1997. Springer.Google Scholar
2. [2]
T. Biedl, B. Madden, and I.G. Tollis. The three-phase method: A unified approach to orthogonal graph drawing. In Di Battista [5], pages 391–402.Google Scholar
3. [3]
T. Biedl, T. Shermer, S. Whitesides, and S. Wismath. Orthogonal 3-D graph drawing. In Di Battista [5], pages 76–86.Google Scholar
4. [4]
T.C. Biedl. Three approaches to 3D-orthogonal box-drawings. In Whitesides [20], pages 30–43.Google Scholar
5. [5]
G. Di Battista, editor. Proc. Graph Drawing: 5th International Symp. (GD’97), volume 1353 of Lecture Notes in Comput. Sci., Berlin, 1998. Springer.Google Scholar
6. [6]
G. Di Battista, W. Didimo, M. Patrignani, and M. Pizzonia. Orthogonal and quasi-upward drawings with vertices of arbitrary size. In J. Kratochvil, editor, Proc. Graph Drawing: 7th International Symp. (GD’99), Lecture Notes in Comput. Sci., Berlin. Springer. to appear.Google Scholar
7. [7]
G. Di Battista, P. Eades, R. Tamassia, and I.G. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, New Jersey, 1999.
8. [8]
G. Di Battista, A. Garg, G. Liotta, R. Tamassia, E. Tassinari, and F. Vargiu. An experimental comparison of four graph drawing algorithms. Comput. Geom., 7(5-6):303–325, 1997.
9. [9]
W. Didimo and G. Liotta. Computing orthogonal drawings in a variable embedding setting. In K.-Y. Chwa and O. H. Ibarra, editors, Proc. Algorithms and Computation (ISAAC’98), volume 1533 of Lecture Notes in Comput. Sci., pages 79–88, Berlin, 1998. Springer.Google Scholar
10. [10]
P. Eades, A. Symvonis, and S. Whitesides. Two algorithms for three dimensional orthogonal graph drawing. In North [16], pages 139–154.Google Scholar
11. [11]
P. Eades, A. Symvonis, and S. Whitesides. Three dimensional orthogonal graph drawing algorithms. 1998. submitted.Google Scholar
12. [12]
S. Even and G. Granot. Grid layouts of block diagrams — bounding the number of bends in each connection. In R. Tamassia and I.G. Tollis, editors, Proc. Graph Drawing: DIMACS International Workshop (GD’94), volume 894 of Lecture Notes in Comput. Sci., pages 64–75, Berlin, 1995. Springer.Google Scholar
13. [13]
U. Fömeier, G. Kant, and M. Kaufmann. 2-visibility drawings of planar graphs. In North [16], pages 155–158.Google Scholar
14. [14]
U. Fömeier and M. Kaufmann. Drawing high degree graphs with low bend numbers. In F.J. Brandenburg, editor, Proc. Graph Drawing: Symp. on Graph Drawing (GD’95), volume 1027 of Lecture Notes in Comput. Sci., pages 254–266, Berlin, 1996. Springer.Google Scholar
15. [15]
U. Fömeier and M. Kaufmann. Algorithms and area bounds for nonplanar orthogonal drawings. In Di Battista [5], pages 134–145.Google Scholar
16. [16]
S. North, editor. Proc. Graph Drawing: Symp. on Graph Drawing (GD’96), volume 1190 of Lecture Notes in Comput. Sci., Berlin, 1997. Springer.Google Scholar
17. [17]
A. Papakostas and I.G. Tollis. Orthogonal drawing of high degree graphs with small area and few bends. In F. Dehne, A. Rau-Chaplin, J.-R. Sack, and R. Tamassia, editors, Proc. Algorithms and Data Structures: 5th International Workshop (WADS’97), volume 1272 of Lecture Notes in Comput. Sci., pages 354–367, Berlin, 1997. Springer.Google Scholar
18. [18]
A. Papakostas and I.G. Tollis. Incremental orthogonal graph drawing in three dimensions. In Di Battista [5], pages 52–63.Google Scholar
19. [19]
R. Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput., 16(3):421–443, June 1987.
20. [20]
S. Whitesides, editor. Proc. Graph Drawing: 6th International Symp. (GD’98), volume 1547 of Lecture Notes in Comput. Sci. Springer, 1998.Google Scholar
21. [21]
D.R. Wood. An algorithm for three-dimensional orthogonal graph drawing. In Whitesides [20], pages 332–346.Google Scholar
22. [22]
D.R. Wood. Multi-dimensional orthogonal graph drawing in the general position model. Technical Report 99/38, School of Computer Science and Software Engineering, Monash University, Australia, 1999. (available at http://www.csse.monash.edu.au/publications/).
23. [23]
D.R. Wood. A new algorithm and open problems in three-dimensional orthogonal graph drawing. In R. Raman and J. Simpson, editors, Proc. Australasian Workshop on Combinatorial Algorithms (AWOCA’99), pages 157–167. Curtin University of Technology, Perth, 1999.Google Scholar