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Drawing Colored Graphs with Constrained Vertex Positions and Few Bends per Edge

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Abstract

Hamiltonicity, book embeddability, and point-set embeddability of planar graphs are strictly related concepts. We exploit the interplay between these notions to describe colored sets of points and to design polynomial-time algorithms to embed k-colored planar graphs on these sets such that the resulting drawings have \(\mathcal{O}(k)\) bends per edge.

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References

  1. Badent, M., Di Giacomo, E., Liotta, G.: Drawing colored graphs on colored points. In: Dehne, F.K.H.A., Sack, J.-R., Zeh, N. (eds.) 10th International Workshop on Algorithms and Data Structures. Lecture Notes in Computer Science, vol. 4619, pp. 102–113. Springer, Berlin (2007). To appear in Theor. Comput. Sci.

    Chapter  Google Scholar 

  2. Batini, C., Talamo, M., Tamassia, R.: Computer aided layout of entity-relationship diagrams. J. Syst. Softw. 4, 163–173 (1984)

    Article  Google Scholar 

  3. Brandes, U., Kenis, P., Wagner, D.: Communicating centrality in policy network drawings. IEEE Trans. Vis. Comput. Graph. 9(2), 241–253 (2003)

    Article  Google Scholar 

  4. Brandes, U., Wagner, D.: Visone—analysis and visualization of social networks. In: Jünger, M., Mutzel, P. (eds.) Graph Drawing Software, pp. 321–340. Springer, Berlin (2004)

    Google Scholar 

  5. Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice-Hall, Upper Saddle River (1999)

    MATH  Google Scholar 

  6. Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H., Trotta, F., Wismath, S.K.: k-colored point-set embeddability of outerplanar graphs. J. Graph Algorithms Appl. 12(1), 29–49 (2008)

    MATH  MathSciNet  Google Scholar 

  7. Di Giacomo, E., Didimo, W., Liotta, G., Wismath, S.K.: Curve-constrained drawings of planar graphs. Comput. Geom. 30, 1–23 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Di Giacomo, E., Didimo, W., Liotta, G., Wismath, S.K.: Book embeddability of series-parallel digraphs. Algorithmica 45(4), 531–547 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Di Giacomo, E., Liotta, G., Trotta, F.: On embedding a graph on two sets of points. IJFCS 17(5), 1071–1094 (2006). Special Issue on Graph Drawing

    MATH  MathSciNet  Google Scholar 

  10. Dorogstev, S.N., Mendes, J.F.F.: Evolution of Networks, From Biological Nets to the Internet and www. Oxford University Press, Oxford (2003)

    Google Scholar 

  11. Eichelberger, H.: Aesthetics of class diagrams. In: Proceedings of the First IEEE International Workshop on Visualizing Software for Understanding and Analysis, pp. 23–31. IEEE, New York (2002)

    Chapter  Google Scholar 

  12. Eichelberger, H., von Gudenberg, J.W.: UML class diagrams—state of the art in layout techniques. In: Proceedings of Vissoft 2003, International Workshop on Visualizing Software for Understanding and Analysis, pp. 30–34 (2003)

  13. Eiglsperger, M., Gutwenger, C., Kaufmann, M., Kupke, J., Jünger, M., Leipert, S., Klein, K., Mutzel, P., Siebenhaller, M.: Automatic layout of UML class diagrams in orthogonal style. Inf. Vis. 3(3), 189–208 (2004)

    Article  Google Scholar 

  14. Enomoto, H., Miyauchi, M.S.: Embedding graphs into a three page book with O(mlog n) crossings of edges over the spine. SIAM J. Discrete Math. 12(3), 337–341 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. Giordano, F., Liotta, G., Mchedlidze, T., Symvonis, A.: Computing upward topological book embeddings of upward planar digraphs. In: Tokuyama, T. (ed.) 18th International Symposium on Algorithms and Computation. Lecture Notes in Computer Science, vol. 4835, pp. 172–183. Springer, Berlin (2007)

    Google Scholar 

  16. Halton, J.H.: On the thickness of graphs of given degree. Inf. Sci. 54, 219–238 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kaneko, A., Kano, M.: Discrete geometry on red and blue points in the plane—a survey. In: Discrete & Computational Geometry. Algorithms and Combinatorics, vol. 25, pp. 551–570. Springer, Berlin (2003)

    Google Scholar 

  18. Kaufmann, M., Wagner, D. (eds.): Drawing Graphs. Lecture Notes in Computer Science, vol. 2025. Springer, Berlin (2001)

    MATH  Google Scholar 

  19. Kaufmann, M., Wiese, R.: Embedding vertices at points: Few bends suffice for planar graphs. J. Graph Algorithms Appl. 6(1), 115–129 (2002)

    MATH  MathSciNet  Google Scholar 

  20. Nishizeki, T., Rahman, M.S. (eds.): Planar Graph Drawing. Lecture Notes Series on Computing, vol. 12. World Scientific, Singapore (2004)

    MATH  Google Scholar 

  21. Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graph Comb. 17, 717–728 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  22. Purchase, H.C.: Which aesthetic has the greatest effect on human understanding? In: Procceedings of GD ’97. Lecture Notes in Computer Science, vol. 1353, pp. 248–261. Springer, Berlin (1998)

    Google Scholar 

  23. Purchase, H.C.: Effective information visualisation: a study of graph drawing aesthetics and algorithms. Interact. Comput. 13(2), 147–162 (2000)

    Article  Google Scholar 

  24. Siebenhaller, M.: Partitioned drawings. In: Kaufmann, M., Wagner, D. (eds.) Graph Drawing. Lecture Notes in Computer Science, vol. 4372, pp. 252–257. Springer, Berlin (2006)

    Chapter  Google Scholar 

  25. Sugiyama, K.: Graph Drawing and Applications for Software and Knowledge Engineers. Word Scientific, Singapore (2002)

    MATH  Google Scholar 

  26. Tamassia, R., Di Battista, G., Batini, C.: Automatic graph drawing and readability of diagrams. IEEE Trans. Syst. Man. Cybern. 18(1), 61–79 (1988)

    Article  Google Scholar 

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Authors and Affiliations

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

    Emilio Di Giacomo, Giuseppe Liotta & Francesco Trotta

Authors
  1. Emilio Di Giacomo
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  2. Giuseppe Liotta
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  3. Francesco Trotta
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Corresponding author

Correspondence to Emilio Di Giacomo.

Additional information

An extended abstract of this paper was presented at the 15th International Symposium on Graph Drawing (GD 2007). Research partially supported by the MIUR Project “MAINSTREAM: Algorithms for massive information structures and data streams”.

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Di Giacomo, E., Liotta, G. & Trotta, F. Drawing Colored Graphs with Constrained Vertex Positions and Few Bends per Edge. Algorithmica 57, 796–818 (2010). https://doi.org/10.1007/s00453-008-9255-2

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