C-Planarity of Extrovert Clustered Graphs

  • Michael T. Goodrich
  • George S. Lueker
  • Jonathan Z. Sun
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3843)

Abstract

A clustered graph has its vertices grouped into clusters in a hierarchical way via subset inclusion, thereby imposing a tree structure on the clustering relationship. The c-planarity problem is to determine if such a graph can be drawn in a planar way, with clusters drawn as nested regions and with each edge (drawn as a curve between vertex points) crossing the boundary of each region at most once. Unfortunately, as with the graph isomorphism problem, it is open as to whether the c-planarity problem is NP-complete or in P. In this paper, we show how to solve the c-planarity problem in polynomial time for a new class of clustered graphs, which we call extrovert clustered graphs. This class is quite natural (we argue that it captures many clustering relationships that are likely to arise in practice) and includes the clustered graphs tested in previous work by Dahlhaus, as well as Feng, Eades, and Cohen. Interestingly, this class of graphs does not include, nor is it included by, a class studied recently by Gutwenger et al.; therefore, this paper offers an alternative advancement in our understanding of the efficient drawability of clustered graphs in a planar way. Our testing algorithm runs in O(n3) time and implies an embedding algorithm with the same time complexity.

References

  1. 1.
    Battista, G.D., Didimo, W., Marcandalli, A.: Planarization of clustered graphs. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 60–74. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Battista, G.D., Tamassia, R.: On-line planarity testing. SIAM J. Comput. 25(5), 956–997 (1996)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Booth, K., Lueker, G.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Systems Sci. 13(3), 335–379 (1976)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Cortese, P.G., Battista, G.D., Patrignani, M., Pizzonia, M.: Clustering cycles into cycles of clusters. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 100–110. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  5. 5.
    Dahlhaus, E.: A linear time algorithm to recognize clustered planar graphs and its parallelization. In: Lucchesi, C.L., Moura, A.V. (eds.) LATIN 1998. LNCS, vol. 1380, pp. 239–248. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  6. 6.
    Duncan, C.A., Goodrich, M.T., Kobourov, S.G.: Planarity-preserving clustering and embedding for large planar graphs. In: Kratochvíl, J. (ed.) GD 1999. LNCS, vol. 1731, pp. 186–196. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  7. 7.
    Eades, P., Feng, Q., Nagamochi, H.: Drawing clustered graphs on an orthogonal grid. Journal of Graph Algorithms and Applications 3(4), 3–29 (1999)MATHMathSciNetGoogle Scholar
  8. 8.
    Eades, P., Feng, Q.-W.: Multilevel visualization of clustered graphs. In: North, S.C. (ed.) GD 1996. LNCS, vol. 1190, pp. 101–112. Springer, Heidelberg (1997)Google Scholar
  9. 9.
    Eades, P., Feng, Q.-W., Lin, X.: Straight-line drawing algorithms for hierarchical graphs and clustered graphs. In: North, S.C. (ed.) GD 1996. LNCS, vol. 1190, pp. 113–128. Springer, Heidelberg (1997)Google Scholar
  10. 10.
    Eades, P., Huang, M.L.: Navigating clustered graphs using force-directed methods. J. Graph Algorithms and Applications: Special Issue on Selected Papers from 1998 Symp. Graph Drawing 4(3), 157–181 (2000)MATHGoogle Scholar
  11. 11.
    Even, S., Tarjan, R.E.: Computing an st-numbering. Theoretical Computer Science 2(3), 339–344 (1976)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Feng, Q.-W., Eades, P., Cohen, R.F.: Clustered graphs and C-planarity. In: Spirakis, P.G. (ed.) ESA 1995. LNCS, vol. 979, pp. 213–226. Springer, Heidelberg (1995)Google Scholar
  13. 13.
    Gutwenger, C., Jünger, M., Leipert, S., Mutzel, P., Percan, M., Weiskircher, R.: Advances in C-planarity testing of clustered graphs. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 220–235. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  14. 14.
    Hopcroft, J., Tarjan, R.E.: Efficient planarity testing. J. ACM 21(4), 549–568 (1974)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hsu, W., McConnell, R.M.: PC trees and circular-ones arrangements. Theoretical Computer Science 296(1), 59–74 (2003)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Lempel, A., Even, S., Cederbaum, I.: An algorithm for planarity testing of graphs. In: Theory of graphs: International symposium, pp. 215–232 (1966)Google Scholar
  17. 17.
    Nishizeki, T., Chiba, N.: Planar Graphs: Theory and Algorithms. Ann. Discrete Math., vol. 32. North-Holland, Amsterdam (1988)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Michael T. Goodrich
    • 1
  • George S. Lueker
    • 1
  • Jonathan Z. Sun
    • 1
  1. 1.Department of Computer Science, Donald Bren School of Information and Computer SciencesUniversity of CaliforniaIrvineUSA

Personalised recommendations