Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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UPRIGHT DRAWINGS OF GRAPHS ON THREE LAYERS

  • Alam, Muhammad Jawaherul (Graph Drawing Research Group, Department of CSE, Bangladesh University of Engineering and Technology (BUET)) ;
  • Rabbi, Md. Mashfiqui (Department of CS, Dartmouth college) ;
  • Rahman, Md. Saidur (Graph Drawing Research Group, Department of CSE, Bangladesh University of Engineering and Technology (BUET)) ;
  • Karim, Md. Rezaul (Department of CSE, University of Dhaka)
  • Received : 2010.01.29
  • Accepted : 2010.03.04
  • Published : 2010.09.30
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Abstract

An upright drawing of a planar graph G on k layers is a planar straight-line drawing of G, where the vertices of G are placed on a set of k horizontal lines, called layers and no two adjacent vertices are placed on the same layer. There is a previously known algorithm that decides in linear time whether a planar graph admits an upright drawing on k layers for a fixed value of k. However, the constant factor in the running time of the algorithm increases exponentially with k and makes it impractical even for k = 3. In this paper, we give a linear-time algorithm to examine whether a biconnected planar graph G admits an upright drawing on three layers and to obtain such a drawing if it exists. We also give a necessary and sufficient condition for a tree to have an upright drawing on three layers. Our algorithms in both the cases are much simpler and easier to implement than the previously known algorithms.

Keywords

References

  1. T. C. Biedl, Drawing planar partitions I: LL-drawings and LH-drawings, Proceedings of the fourteenth annual symposium on Computational geometry, (1998), 287-296.
  2. S. Cornelsen, T. Schank, and D. Wagner, Drawing graphs on two and three lines, Journal of Graph Algorithms and Applications, 8(2), (2004), 161-177. https://doi.org/10.7155/jgaa.00087
  3. G. Di Battista, P. Eades, R. Tamassia, and I. G. Tollis, Graph Drawing: Algorithms for the Visualization of Graphs, Prentice-Hall, Upper Saddle River, New Jersey, 1999.
  4. V. Dujmovic, M. R. Fellows, M. T. Hallett, M. Kitching, G. Liotta, C. McCartin, N. Nishimura, P. Ragde, F. A. Rosamond, M. Suderman, S. Whitesides, and D. R. Wood, On the parameterized complexity of layered graph drawing, Proceedings of the 9th Annual European Symposium on Algorithms (ESA), 2161, Lecture Notes in Computer Science, Springer-Verlag, (2001), 488-499.
  5. U. Fossmeier and M. Kaufmann, Nice drawings for planar bipartite graphs, Proceedings of the 3rd Italian Conference on Algorithms and Complexity, Springer-Verlag, (1997), 122-134.
  6. S. Felsner, G. Liotta, and S. K. Wismath, Straight-line drawings on restricted integer grids in two and three dimensions, Journal of Graph Algorithms and Applications, 7(4), (2003), 363-398. https://doi.org/10.7155/jgaa.00075
  7. M. Kaufmann and D. Wagner, Drawing Graphs: Methods and Models, 2025, Lecture Notes in Computer Science, Springer-Verlag, London, UK, 2001.
  8. T. Lengauer, Combinatorial Algorithms for Integrated Circuit Layouts, Wiley, New York, NY, USA, 1990.
  9. P. Scheffler, A linear algorithm for the pathwidth of trees, Topics in Combinatorics and Graph Theory, Physica-Verlag, (1990), 613-620.
  10. M. Suderman, Pathwidth and layered drawing of trees, International Journal of Computational Geometry and Applications, 14(3), (2004), 203-225. https://doi.org/10.1142/S0218195904001433
  11. M. Suderman, Proper and planar drawings of graphs on three layers, Proceedings of the 15th International Symposium on Graph Drawing, 3843, Lecture Notes in Computer Science, Springer-Verlag, (2006), 434-445.
  12. M. S. Waterman and J. R. Griggs, Interval graphs and maps of DNA, Bulletin of Mathematical Biology, 48, (1986), 189-195.