Advertisement

Discrete & Computational Geometry

, Volume 45, Issue 1, pp 88–140 | Cite as

Straight-Line Rectangular Drawings of Clustered Graphs

  • Patrizio Angelini
  • Fabrizio FratiEmail author
  • Michael Kaufmann
Article

Abstract

We show that every c-planar clustered graph has a straight-line c-planar drawing in which each cluster is represented by an axis-parallel rectangle, thus solving a problem posed by Eades, Feng, Lin, and Nagamochi (Algorithmica 44(1):1–32, 2006).

Keywords

Clustered graphs Planar graphs Straight-line drawings Rectangles 

References

  1. 1.
    Cornelsen, S., Wagner, D.: Completely connected clustered graphs. J. Discrete Algorithms 4(2), 313–323 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cortese, P.F., Di Battista, G., Frati, F., Patrignani, M., Pizzonia, M.: C-planarity of c-connected clustered graphs. J. Graph Algorithms Appl. 12(2), 225–262 (2008) zbMATHMathSciNetGoogle Scholar
  3. 3.
    Cortese, P.F., Di Battista, G., Patrignani, M., Pizzonia, M.: Clustering cycles into cycles of clusters. J. Graph Algorithms Appl. 9(3), 391–413 (2005) zbMATHMathSciNetGoogle Scholar
  4. 4.
    Cortese, P.F., Di Battista, G., Patrignani, M., Pizzonia, M.: On embedding a cycle in a plane graph. Discrete Math. 309(7), 1856–1869 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dahlhaus, E.: A linear time algorithm to recognize clustered graphs and its parallelization. In: Lucchesi, C.L., Moura, A.V. (eds.) Latin American Symposium on Theoretical Informatics (LATIN ’98), pp. 239–248 (1998) CrossRefGoogle Scholar
  6. 6.
    Di Battista, G., Drovandi, G., Frati, F.: How to draw a clustered tree. J. Discrete Algorithms 7(4), 479–499 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice Hall, Upper Saddle River (1999) zbMATHGoogle Scholar
  8. 8.
    Di Battista, G., Frati, F.: Efficient c-planarity testing for embedded flat clustered graphs with small faces. J. Graph Algorithms Appl. 13(3), 349–378 (2009) zbMATHMathSciNetGoogle Scholar
  9. 9.
    Di Battista, G., Frati, F., Patrignani, M.: Non-convex representations of graphs. In: Tollis, I., Patrignani, M. (eds.) Graph Drawing (GD ’08), pp. 390–395 (2008) Google Scholar
  10. 10.
    Di Battista, G., Tamassia, R.: Algorithms for plane representations of acyclic digraphs. Theor. Comput. Sci. 61, 175–198 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Eades, P., Feng, Q., Lin, X., Nagamochi, H.: Straight-line drawing algorithms for hierarchical graphs and clustered graphs. Algorithmica 44(1), 1–32 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Eades, P., Feng, Q., Nagamochi, H.: Drawing clustered graphs on an orthogonal grid. J. Graph Algorithms Appl. 3(4), 3–29 (1999) zbMATHMathSciNetGoogle Scholar
  13. 13.
    Eades, P., Feng, Q.W., Lin, X.: Straight-line drawing algorithms for hierarchical graphs and clustered graphs. In: North, S. (ed.) Graph Drawing (GD ’96), pp. 113–128 (1996) Google Scholar
  14. 14.
    Fáry, I.: On straight line representations of planar graphs. Acta Sci. Math. 11, 229–233 (1948) Google Scholar
  15. 15.
    Feng, Q.: Algorithms for drawing clustered graphs. Ph.D. Thesis, The University of Newcastle, Australia (1997) Google Scholar
  16. 16.
    Feng, Q., Cohen, R.F., Eades, P.: How to draw a planar clustered graph. In: Du, D., Li, M. (eds.) Computing and Combinatorics (COCOON ’95), pp. 21–30 (1995) CrossRefGoogle Scholar
  17. 17.
    Feng, Q., Cohen, R.F., Eades, P.: Planarity for clustered graphs. In: Spirakis, P.G. (ed.) European Symposium on Algorithms (ESA ’95), pp. 213–226 (1995) Google Scholar
  18. 18.
    Goodrich, M.T., Lueker, G.S., Sun, J.Z.: C-planarity of extrovert clustered graphs. In: Healy, P., Nikolov, N.S. (eds.) Graph Drawing (GD ’05), pp. 211–222 (2005) Google Scholar
  19. 19.
    Gutwenger, C., Jünger, M., Leipert, S., Mutzel, P., Percan, M., Weiskircher, R.: Advances in c-planarity testing of clustered graphs. In: Kobourov, S.G., Goodrich, M.T. (eds.) Graph Drawing (GD ’02), pp. 220–235 (2002) CrossRefGoogle Scholar
  20. 20.
    Jelínek, V., Jelínková, E., Kratochvíl, J., Lidický, B.: Clustered planarity: Embedded clustered graphs with two-component clusters. In: Tollis, I.G., Patrignani, M. (eds.) Graph Drawing (GD ’08), pp. 121–132 (2008) Google Scholar
  21. 21.
    Jelínek, V., Suchý, O., Tesar, M., Vyskocil, T.: Clustered planarity: Clusters with few outgoing edges. In: Tollis, I.G., Patrignani, M. (eds.) Graph Drawing (GD ’08), pp. 102–113 (2008) Google Scholar
  22. 22.
    Jelínková, E., Kára, J., Kratochvíl, J., Pergel, M., Suchý, O., Vyskocil, T.: Clustered planarity: Small clusters in cycles and eulerian graphs. J. Graph Algorithms Appl. 13(3), 379–422 (2009) zbMATHMathSciNetGoogle Scholar
  23. 23.
    Jünger, M., Leipert, S., Percan, M.: Triangulating clustered graphs. Technical Report, Zentrum für Angewandte Informatik Köln, Lehrstuhl Jünger, December 2002 Google Scholar
  24. 24.
    Kaufmann, M., Wagner, D. (eds.): Drawing Graphs, Methods and Models. Lecture Notes in Computer Science. Springer, Berlin (2001) zbMATHGoogle Scholar
  25. 25.
    Nagamochi, H., Kuroya, K.: Drawing c-planar biconnected clustered graphs. Discrete Appl. Math. 155(9), 1155–1174 (2007) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Patrizio Angelini
    • 1
  • Fabrizio Frati
    • 1
    Email author
  • Michael Kaufmann
    • 2
  1. 1.Departimento di Informatica e AutomazioneRoma Tre UniversityRomeItaly
  2. 2.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenGermany

Personalised recommendations