# Path-Width and Three-Dimensional Straight-Line Grid Drawings of Graphs

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## Abstract

We prove that every n-vertex graph *G* with path-width **pw**(*G*) has a three-dimensional straight-line grid drawing with *O*(**pw**(*G*)^{2}·*n*) volume. Thus for graphs with bounded path-width the volume is *O*(*n*), and it follows that for graphs with bounded tree-width, such as series-parallel graphs, the volume is *O*(*n* log^{2}*n*). No better bound than *O*(*n*^{2}) was previously known for drawings of series-parallel graphs. For planar graphs we obtain three-dimensional drawings with *O*(*n*^{2}) volume and *O(√n)* aspect ratio, whereas all previous constructions with *O*(*n*^{2}) volume have *Θ(n)* aspect ratio.

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### References

- [1]H. L. Bodlaender, A linear-time algorithm for finding tree-decompositions of small treewidth.
*SIAM J. Comput.*,**25(6)**:1305–1317, 1996.MATHCrossRefMathSciNetGoogle Scholar - [2]H. L. Bodlaender, A partial
*k*-arboretum of graphs with bounded treewidth.*Theoret. Comput. Sci.*,**209(1–2)**:1–45, 1998.MATHCrossRefMathSciNetGoogle Scholar - [3]F. J. Brandenburg, ed.,
*Proc. International Symp. on Graph Drawing (GD’ 95)*, vol. 1027 of*Lecture Notes in Comput. Sci.*, Springer, 1996.Google Scholar - [4]
- [5]T. Calamoneri and A. Sterbini, 3D straight-line grid drawing of 4-colorable graphs.
*Inform. Process. Lett.*,**63(2)**:97–102, 1997.CrossRefMathSciNetGoogle Scholar - [6]K. Chilakamarri, N. Dean, and M. Littman, Three-dimensional Tutte embedding. In
*Proc. 26th Southeastern International Conf. on Combinatorics, Graph Theory and Computing*, vol. 107 of*Cong. Numer.*, pp. 129–140, 1995.Google Scholar - [7]M. Chrobak, M. Goodrich, and R. Tamassia, Convex drawings of graphs in two and three dimensions. In
*Proc. 12th Annual ACM Symp. on Comput. Geom.*, pp. 319–328, 1996.Google Scholar - [8]R. F. Cohen, P. Eades, T. Lin, and F. Ruskey, Three-dimensional graph drawing.
*Algorithmica*,**17(2)**:199–208, 1996CrossRefMathSciNetGoogle Scholar - [9]I. F. Cruz and J. P. Twarog, 3D graph drawing with simulated annealing. In [3], pp. 162–165.Google Scholar
- [10]H. de Fraysseix, J. Pach, and R. Pollack, How to draw a planar graph on a grid.
*Combinatorica*,**10(1)**:41–51, 1990.MATHCrossRefMathSciNetGoogle Scholar - [11]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 - [12]
- [13]V. Dujmović, M. Fellows, M. Hallett, M. Kitching, G. Liotta, C. McCartin, N. Nishimura, P. Ragde, F. Rosemand, M. Suderman, S. Whitesides, and D. R. Wood, On the parameterized complexity of layered graph drawing. In F. Meyer auf der Heide, ed.,
*Proc. 5th Annual European Symp. on Algorithms (ESA’ 01)*, vol. 2161 of*Lecture Notes in Comput. Sci.*, pp. 488–499, Springer, 2001.Google Scholar - [14]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
- [15]P. Eades and P. Garvan, Drawing stressed planar graphs in three dimensions. In [3], pp. 212–223.Google Scholar
- [16]P. Erdös, Appendix. In K. F. Roth, On a problem of Heilbronn.
*J. London Math. Soc.*,**26**:198–204, 1951.Google Scholar - [17]S. Felsner, S. Wismath, and G. Liotta, Straight-line drawings on restricted integer grids in two and three dimensions. In [28], pp. 328–342.Google Scholar
- [18]A. Garg, R. Tamassia, and P. Vocca, Drawing with colors. In J. Diaz and M. Serna, eds.,
*Proc. 4th Annual European Symp. on Algorithms (ESA’ 96)*, vol. 1136 of*Lecture Notes in Comput. Sci.*, pp. 12–26, SpringerGoogle Scholar - [19]A. Gupta and N. Nishimura, Sequential and parallel algorithms for embedding problems on classes of partial
*k*-trees. In*Proc. 4th Scandinavian Workshop on Algorithm Theory (SWAT’ 94)*, vol. 824 of*Lecture Notes in Comput. Sci.*, pp. 172–182, Springer, 1984.Google Scholar - [20]A. Gupta, N. Nishimura, A. Proskurowski, and P. Ragde, Embeddings of k-connected graphs of pathwidth
*k*. In M. M. Halldorsson, ed.,*Proc. 7th Scandinavian Workshop on Algorithm Theory (SWAT’ 00)*, vol. 1851 of*Lecture Notes in Comput. Sci.*, pp. 111–124, Springer, 2000.Google Scholar - [21]P. Hliněný, Crossing-critical graphs and path-width. In [28], pp. 102–114.Google Scholar
- [22]S.-H. Hong, Drawing graphs symmetrically in three dimensions. In [28], pp. 189–204.Google Scholar
- [23]S.-H. Hong and P. Eades, An algorithm for finding three dimensional symmetry in series parallel digraphs. In D. Lee and S.-H. Teng, eds.,
*Proc. 11th International Conf. on Algorithms and Computation (ISAAC’ 00)*, vol. 1969 of*Lecture Notes in Comput. Sci.*, pp. 266–277, Springer, 2000.Google Scholar - [24]S.-H. Hong and P. Eades, An algorithm for finding three dimensional symmetry in trees. In J. Marks, ed.,
*Proc. 8th International Symp. on Graph Drawing (GD’ 00)*, vol. 1984 of*Lecture Notes in Comput. Sci.*, pp. 360–371, Springer, 2001.Google Scholar - [25]S.-H. Hong, P. Eades, A. Quigley, and S.-H. Lee, Drawing algorithms for series-parallel digraphs in two and three dimensions. In S. Whitesides, ed.,
*Proc. 6th International Symp. on Graph Drawing (GD’ 98)*, vol. 1547 of*Lecture Notes in Comput. Sci.*, pp. 198–209, Springer, 1998.Google Scholar - [26]F. T. Leighton and A. L. Rosenberg, Three-dimensional circuit layouts.
*SIAM J. Comput.*,**15(3)**:793–813, 1986.MATHCrossRefMathSciNetGoogle Scholar - [27]B. Monien, F. Ramme, and H. Salmen, A parallel simulated annealing algorithm for generating 3D layouts of undirected graphs. In [3], pp. 396–408.Google Scholar
- [28]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.*, Springer, 2002.Google Scholar - [29]D. I. Ostry,
*Some Three-Dimensional Graph Drawing Algorithms*. Master’s thesis, Department of Computer Science and Software Engineering, The University of Newcastle, Australia, 1996.Google Scholar - [30]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 - [31]Z. Peng,
*Drawing Graphs of Bounded Treewidth/Pathwidth*. Master’s thesis, Department of Computer Science, University of Auckland, New Zealand, 2001.Google Scholar - [32]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
- [33]W. Schnyder, Planar graphs and poset dimension.
*Order*,**5(4)**:323–343, 1989.MATHCrossRefMathSciNetGoogle Scholar - [34]M. Thorup, All structured programs have small tree-width and good register allocation.
*Information and Computation*,**142(2)**:159–181, 1998.MATHCrossRefMathSciNetGoogle Scholar - [35]C. Ware and G. Franck, Viewing a graph in a virtual reality display is three times as good as a 2D diagram. In A. L. Ambler and T. D. Kimura, eds.,
*Proc. IEEE Symp. Visual Languages (VL’ 94)*, pp. 182–183, IEEE, 1994.Google Scholar - [36]C. Ware, D. Hui, and G. Franck, Visualizing object oriented software in three dimensions. In
*Proc. IBM Centre for Advanced Studies Conf. (CASCON’ 93)*, pp. 1–11, 1993.Google Scholar - [37]R. Weiskircher, Drawing planar graphs. In M. Kaufmann and D. Wagner, eds.,
*Drawing Graphs: Methods and Models*, vol. 2025 of*Lecture Notes in Comput. Sci.*, pp. 23–45, Springer, 2001.Google Scholar - [38]D. R. Wood, Queue layouts, tree-width, and three-dimensional graph drawing. Tech. Rep. TR-02-02 (revised), School of Computer Science, Carleton University, Ottawa, Canada, August, 2002.Google Scholar

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© Springer-Verlag Berlin Heidelberg 2002