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On the Parameterized Complexity of Layered Graph Drawing

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Abstract

We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer drawing (for fixed h). This algorithm is extended to solve related problems, including allowing at most k crossings, or removing at most r edges to leave a crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems are NP-complete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case either the total span or the maximum span of edges can be minimized. In contrast to the Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.

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References

  1. Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)

    Article  MATH  Google Scholar 

  2. Buchheim, C., Jünger, M., Leipert, S.: A fast layout algorithm for k-level graphs. In: Marks, J. (ed.) Proc. Graph Drawing: 8th Internat. Symp. (GD’00). Lecture Notes in Computer Science, vol. 1984, pp. 229–240. Springer, New York (2001)

    Google Scholar 

  3. Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21, 358–402 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bienstock, D., Monma, C.L.: On the complexity of embedding planar graphs to minimize certain distance measures. Algorithmics 5, 93–109 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  6. Carpano, M.J.: Automatic display of hierarchized graphs for computer aided decision analysis. IEEE Trans. Syst. Man Cybern. 10(11), 705–715 (1980)

    Article  Google Scholar 

  7. Cornelsen, S., Schank, T., Wagner, D.: Drawing graphs on two and three lines. J. Graph. Algorithms Appl. 8(2), 161–177 (2004)

    MATH  MathSciNet  Google Scholar 

  8. Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, New York (1999)

    MATH  Google Scholar 

  9. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)

    Google Scholar 

  10. Dujmović, V., Fellows, M., Hallett, M., Kitching, M., Liotta, G., McCartin, C., Nishimura, N., Ragde, P., Rosamond, F., Suderman, M., Whitesides, S., Wood, D.R.: On the parameterized complexity of layered graph drawing. In: Meyer auf der Heide, F. (ed.) European Symposium on Algorithms, pp. 488–499 (2001)

  11. Dujmović, V., Fellows, M., Hallett, M., Kitching, M., Liotta, G., McCartin, C., Nishimura, N., Ragde, P., Rosamond, F., Suderman, M., Whitesides, S., Wood, D.R.: A fixed-parameter approach to two-layer planarization. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) Proc. 9th Internat. Symp. on Graph Drawing (GD ’01). Lecture Notes in Computer Science, vol. 2265, pp. 1–15. Springer, New York (2002)

    Google Scholar 

  12. Dujmović, V., Fellows, M., Hallett, M., Kitching, M., Liotta, G., McCartin, C., Nishimura, N., Ragde, P., Rosamond, F., Suderman, M., Whitesides, S., Wood, D.R.: A fixed-parameter approach to two-layer planarization. Algorithmica 45(2), 159–182 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dujmović, V., Fernau, H., Kaufmann, M.: Fixed parameter algorithms for one-sided crossing minimization revisited. J. Discrete Algorithms (2007). doi:10.1016/j.jda.2006.12.008

    Google Scholar 

  14. Diestel, R.: Graph theory, 2nd edn. Graduate Texts in Mathematics, vol. 173. Springer, New York (2000)

    Google Scholar 

  15. Dujmović, V., Morin, P., Wood, D.R.: Layout of graphs with bounded tree-width. SIAM J. Comput. 34(3), 553–579 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dujmović, V., Whitesides, S.: An efficient fixed parameter tractable algorithm for 1-sided crossing minimization. Algorithmica 40(1), 15–31 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Eades, P., Whitesides, S.: Drawing graphs in two layers. Theor. Comput. Sci. 131(2), 361–374 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  18. Eades, P., Wormald, N.C.: Edge crossings in drawings of bipartite graphs. Algorithmica 11(4), 379–403 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fernau, H.: Two-layer planarization: improving on parameterized algorithmics. J. Graph. Algorithms Appl. 9(2), 205–238 (2005)

    MATH  MathSciNet  Google Scholar 

  20. Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM J. Algebr. Discrete Methods 4(3), 312–316 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gansner, E.R., Koutsofios, E., North, S.C., Vo, K.-P.: A technique for drawing directed graphs. IEEE Trans. Softw. Eng. 19(3), 214–230 (1993)

    Article  Google Scholar 

  22. Gupta, A., Nishimura, N., Proskurowski, A., Ragde, P.: Embeddings of k-connected graphs of pathwidth k. Discrete Appl. Math. 145(2), 242–265 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Grohe, M.: Computing crossing numbers in quadratic time. J. Comput. Syst. Sci. 68(2), 285–302 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Heath, L.S., Rosenberg, A.L.: Laying out graphs using queues. SIAM J. Comput. 21(5), 927–958 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kawarabayashi, K., Reed, B.: Computing crossing number in linear time. In: Proc. 39th Annual ACM Symposium on Theory of Computing (STOC ’07), pp. 382–390. ACM Press, New York (2007)

    Chapter  Google Scholar 

  26. Lengauer, T.: Combinatorial Algorithms for Integrated Circuit Layout. Wiley, New York (1990)

    MATH  Google Scholar 

  27. Mutzel, P.: Optimization in leveled graphs. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, vol. 4, pp. 189–196. Kluwer, Dordrecht (2001)

    Google Scholar 

  28. Sander, G.: A fast heuristic for hierarchical Manhattan layout. In: Proc. Internat. Symp. on Graph Drawing (GD ’95). Lecture Notes in Computer Science, vol. 1027, pp. 447–458. Springer, Berlin (1996)

    Chapter  Google Scholar 

  29. Sugiyama, K., Tagawa, S., Toda, M.: Methods for visual understanding of hierarchical system structures. IEEE Trans. Syst. Man Cybern. 11(2), 109–125 (1981)

    Article  MathSciNet  Google Scholar 

  30. Suderman, M.: Pathwidth and layered drawings of trees. Int. J. Comput. Geom. Appl. 14(3), 203–225 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  31. Suderman, M., Whitesides, S.: Experiments with the fixed-parameter approach for two-layer planarization. J. Graph. Algorithms Appl. 9(1), 149–163 (2005)

    MATH  MathSciNet  Google Scholar 

  32. Tomii, N., Kambayashi, Y., Yajima, S.: On planarization algorithms of 2-level graphs. Pap. Tech. Group Electron. Comput. IECEJ 38, 1–12 (1977)

    Google Scholar 

  33. Warfield, J.N.: Crossing theory and hierarchy mapping. IEEE Trans. Syst. Man Cybern. 7(7), 505–523 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  34. Waterman, M.S., Griggs, J.R.: Interval graphs and maps of DNA. Bull. Math. Biol. 48(2), 189–195 (1986)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2T5, Canada

    Vida Dujmović, Matthew Kitching & Sue Whitesides

  2. School of Electrical Engineering and Computer Science, The University of Newcastle, Newcastle, Australia

    Michael R. Fellows & Frances Rosamond

  3. Dipartimento di Ingegneria Elettronica e dell’Informazione, Università degli Studi di Perugia, Perugia, Italy

    Giuseppe Liotta

  4. Institute of Information Science and Technology, Massey University, Palmerston North, New Zealand

    Catherine McCartin

  5. School of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 2P9, Canada

    Naomi Nishimura & Prabhakar Ragde

  6. Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain

    David R. Wood

Authors
  1. Vida Dujmović
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  2. Michael R. Fellows
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  3. Matthew Kitching
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  4. Giuseppe Liotta
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  5. Catherine McCartin
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  6. Naomi Nishimura
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  7. Prabhakar Ragde
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  8. Frances Rosamond
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  9. Sue Whitesides
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  10. David R. Wood
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Corresponding author

Correspondence to Prabhakar Ragde.

Additional information

Research initiated at the International Workshop on Fixed Parameter Tractability in Graph Drawing, Bellairs Research Institute of McGill University, Holetown, Barbados, Feb. 9–16, 2001, organized by S. Whitesides. Research of Canada-based authors is supported by NSERC; research of Quebec-based authors is also supported by a grant from FCAR. Research of D.R. Wood completed while visiting McGill University. Research of G. Liotta supported by CNR and MURST.

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Dujmović, V., Fellows, M.R., Kitching, M. et al. On the Parameterized Complexity of Layered Graph Drawing. Algorithmica 52, 267–292 (2008). https://doi.org/10.1007/s00453-007-9151-1

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  • DOI: https://doi.org/10.1007/s00453-007-9151-1

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