Some Applications of Orderly Spanning Trees in Graph Drawing
Orderly spanning trees seem to have the potential of becoming a new and promising technique capable of unifying known results as well as deriving new results in graph drawing. Our exploration in this paper provides new evidence to demonstrate such a potential. Two applications of the orderly spanning trees of plane graphs are investigated. Our first application deals with Podevs drawing, i.e., planar orthogonal drawing with equal vertex size, introduced by Fößmeier and Kaufmann. Based upon orderly spanning trees, we give an algorithm that produces a Podevs drawing with half-perimeter no more than [3n/2] + 1 and at most one bend per edge for any n-node plane graph with maximal degree Δ, a notable improvement over the existing results in the literature in terms of the size of the drawing area. The second application is an alternative proof for the sufficient and necessary condition for a graph to admit a rectangular dual, i.e., a floor-plan using only rectangles.
KeywordsPlane Graph External Boundary Graph Drawing Plane Triangulation Canonical Ordering
- 2.T. Biedl and M. Kaufmann. Area-efficient static and incremental graph drawings. In Proceedings of the 5th European Symposium on Algorithms, Lecture Notes in Computer Science 1284, pages 37–52. Springer-Verlag, 1997.Google Scholar
- 4.N. Bonichon, B. Le Saëc, and M. Mosbah. Orthogonal drawings based on the stratification of planar graphs. In Proceedings of the 6th International Conference on Graph Theory, Electronic Notes in Discrete Math 5, Marseille, France, 2000. Elsevier. A full version can be found at http://citeseer.nj.nec.com/bonichon00orthogonal.html.
- 5.Y.-T. Chiang, C.-C. Lin, and H.-I. Lu. Orderly spanning trees with applications to graph encoding and graph drawing. In Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 506–515, Washington, D. C., USA, 7–9 Jan. 2001. A revised and extended version can be found at http://xxx.lanl.gov/abs/cs.DS/0102006.
- 10.G. di Battista, P. Eades, R. Tammassia, and I. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, 1998.Google Scholar
- 11.M. Eiglsperger and M. Kaufmann. Fast compaction for orthogonal drawings with vertices of prescribed size. In Mutzel et al. , pages 124–138.Google Scholar
- 12.U. Fö\meier, G. Kant, and M. Kaufmann. 2-visibility drawings of planar graphs. In S. North, editor, Proceedings of the 4th International Symposium on Graph Drawing, Lecture Notes in Computer Science 1190, pages 155–168, California, USA, 1996. Springer-Verlag.Google Scholar
- 13.U. Fößmeier and M. Kaufmann. Drawing high degree graphs with low bend numbers. In S. Whitesides, editor, Proceedings of the 3th International Symposium on Graph Drawing, Lecture Notes in Computer Science 1027, pages 254–266, Passau, Germany, 1995. Springer-Verlag. A full version: Technical Report WSI-95-21, Wilhelm-Schickard-Institut Universität Tübingen, 1995.Google Scholar
- 14.R. L. Francis and J. A. White. Facility layout and location. Prentice-Hall, New Jersey, 1974.Google Scholar
- 17.O. H. Ibarra and L. Zhang, editors. Proceedings of the 8th International Conference on Computing and Combinatorics, Lecture Notes in Computer Science 2387, Singapore, August 15–17 2002. Springer.Google Scholar
- 23.C.-C. Liao, H.-I. Lu, and H.-C. Yen. Floor-planning via orderly spanning trees. In Mutzel et al. , pages 367–377.Google Scholar
- 24.H.-I. Lu. Improved compact routing tables for planar networks via orderly spanning trees. In Ibarra and Zhang , pages 57–66.Google Scholar
- 25.K. Mailing, S. H. Mueller, and W. R. Heller. On finding most optimal rectangular package plans. In Proceedings of the 19th Annual IEEE Design Automation Conference, pages 263–270, 1982.Google Scholar
- 26.M. Mosbah, N. Bonichon, and B. Le Saëc. Wagner’s theorem on realizers. In M. Hennessy and P. Widmayer, editors, Proceedings of the 29th International Colloquium on Automata, Languages, and Programming, Lecture Notes in Computer Science 1443, pages 1043–1063, Málaga, Spain, 2002. Springer-Verlag.Google Scholar
- 27.P. Mutzel, M. Jünger, and S. Leipert, editors. Proceedings of the 9th International Symposium on Graph Drawing, Lecture Notes in Computer Science 2265, Vienna, Austria, 2001. Springer.Google Scholar
- 28.P. Mutzel and R. Weiskircher. Bend minimization in orthogonal drawings using integer programming. In Ibarra and Zhang , pages 484–493.Google Scholar
- 29.R. Otten. Efficient floorplan optimization. In Proceedings of International Conference on Computer Design, pages 499–503, Port Chester, New York, 1983.Google Scholar
- 31.M. S. Rahman, S. Nakano, and T. Nishizeki. Box-rectangular drawings of plane graph. In Proceedings of the 21st Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science 1272, pages 250–261. Springer-Verlag, 1999.Google Scholar
- 32.W. Schnyder. Embedding planar graphs on the grid. In Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 138–148, 1990.Google Scholar
- 33.N. Sherwani. Algorithms for VLSI physical design automation. Kluwer Academic Publishers, 1995.Google Scholar
- 36.S. Tsukiyama, K. Koike, and I. Shirakawa. An algorithm to eliminate all complex triangles in a maximal planar graph for use in VLSI floorplan. In Proceedings of the IEEE International Symposium on Circuits and Systems, pages 321–324, 1986.Google Scholar