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Orthogonal Drawings of Series-Parallel Graphs with Minimum Bends

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Algorithms and Computation (ISAAC 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3827))

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Abstract

In an orthogonal drawing of a planar graph G, each vertex is drawn as a point, each edge is drawn as a sequence of alternate horizontal and vertical line segments, and any two edges do not cross except at their common end. A bend is a point where an edge changes its direction. A drawing of G is called an optimal orthogonal drawing if the number of bends is minimum among all orthogonal drawings of G. In this paper we give an algorithm to find an optimal orthogonal drawing of any given series-parallel graph of the maximum degree at most three. Our algorithm takes linear time, while the previously known best algorithm takes cubic time. Furthermore, our algorithm is much simpler than the previous one. We also obtain a best possible upper bound on the number of bends in an optimal drawing.

This work is supported by JSPS grants.

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References

  1. Bertolazzi, P., Cohen, R.F., Di Battista, G., Tamassia, R., Tollis, I.G.: How to draw a series-parallel digraph. In: Nurmi, O., Ukkonen, E. (eds.) SWAT 1992. LNCS, vol. 621, pp. 272–283. Springer, Heidelberg (1992)

    Google Scholar 

  2. Brandenburg, F., Eppstein, D., Goodrich, M.T., Kobourov, S., Liotta, G., Mutzel, P.: Selected open problems in graph drawings. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 515–539. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  3. Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithm for the Visualization of Graphs. Prentice-Hall Inc., Upper Saddle River (1999)

    Google Scholar 

  4. Di Battista, G., Liotta, G., Vargiu, F.: Spirality and optimal orthogonal drawings. Tech. Rept. 07.94, Dipartimento di Informatica e Sistemistica, Universita’ di Roma La Sapienza (1994)

    Google Scholar 

  5. Di Battista, G., Liotta, G., Vargiu, F.: Spirality and optimal orthogonal drawings. SIAM J. Comput. 27(6), 1764–1811 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Garg, A., Tamassia, R.: A new minimum cost flow algorithm with applications to graph drawing. In: North, S.C. (ed.) GD 1996. LNCS, vol. 1190, pp. 201–226. Springer, Heidelberg (1997)

    Google Scholar 

  7. Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31(2), 601–625 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hong, S.H., Eades, P., Lee, S.H.: Drawig series-parallel digraphs symmetrically. Comput. Geom. 17, 165–188 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Juvan, M., Mohar, B., Thomas, R.: List edge-colorings of series-parallel graphs. The Electronic J. Combinatorics 6(42), 1–6 (1999)

    MathSciNet  Google Scholar 

  10. Nishizeki, T., Rahman, M.S.: Planar Graph Drawing. World Scientific, Singapore (2004)

    MATH  Google Scholar 

  11. Nomura, K., Taya, S., Ueno, S.: On the orthogonal drawing of outerplanar graphs. In: Chwa, K.-Y., Munro, J.I.J. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 300–308. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  12. Rahman, M.S., Egi, N., Nishizeki, T.: No-bend orthogonal drawings of subdivisions of planar triconnected cubic graphs. IEICE Trans. Inf. & Syst. E88-D(1), 23–30 (2005)

    Article  Google Scholar 

  13. Rahman, M.S., Nishizeki, T.: Bend-minimum orthogonal drawings of plane 3-graphs. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 367–378. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  14. Rahman, M.S., Nakano, S., Nishizeki, T.: Rectangular grid drawings of plane graphs. Comp. Geom. Theo. App. 10(3), 203–220 (1998)

    MATH  MathSciNet  Google Scholar 

  15. Rahman, M.S., Nakano, S., Nishizeki, T.: A linear algorithm for bend-optimal orthogonal drawings of triconnected cubic plane graphs. J. of Graph Alg. App. 3(4), 31–62 (1999)

    MATH  MathSciNet  Google Scholar 

  16. Rahman, M.S., Nakano, S., Nishizeki, T.: Box-rectangular drawings of plane graphs. J. of Algorithms 37, 363–398 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Rahman, M.S., Nakano, S., Nishizeki, T.: Orthogonal drawings of plane graphs without bends. J. of Graph Alg. App. 7(4), 335–362 (2003)

    MATH  Google Scholar 

  18. Storer, J.: On minimum node-cost planar embeddings. Networks 14, 181–212 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  19. Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16, 421–444 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  20. Takamizawa, K., Nishizeki, T., Saito, N.: Linear-time computability of combinatorial problems on series-parallel graphs. J. Assoc. Comput. Mach. 29, 623–641 (1982)

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Graduate School of Information Sciences, Tohoku University, Aoba-yama 6-6-05, Sendai, 980-8579, Japan

    Xiao Zhou & Takao Nishizeki

Authors
  1. Xiao Zhou
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  2. Takao Nishizeki
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Editor information

Editors and Affiliations

  1. Department of Computer Science, City University of Hong Kong, Tat Chee Avenue, Kowloon , Hong Kong

    Xiaotie Deng

  2. Department of Computer Science, University of Texas at Dallas, EE/CS Building, 75083, Richardson, TX, USA

    Ding-Zhu Du

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© 2005 Springer-Verlag Berlin Heidelberg

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Zhou, X., Nishizeki, T. (2005). Orthogonal Drawings of Series-Parallel Graphs with Minimum Bends. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_18

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  • DOI: https://doi.org/10.1007/11602613_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-30935-2

  • Online ISBN: 978-3-540-32426-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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