Advertisement

Abstract

Planarity is an important concept in graph drawing. It is generally accepted that planar drawings are well understandable. Recently, several variations of planarity have been studied for advanced graph concepts such as k-level graphs [16, 15, 13, 14, 11, 12, 10, 6] and clustered graphs [7, 5]. In k-level graphs, the vertices are partitioned into k levels and the vertices of one level are drawn on a horizontal line. In clustered graphs, there is a recursive clustering of the vertices according to a given nesting relation. In this paper we combine the concepts of level planarity and clustering and introduce clustered k-level graphs. For connected clustered level graphs we show that clustered k-level planarity can be tested in \(\mathcal O(k|v|)\) time.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Booth, K.S., Lueker, G.S.: Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-Tree Algorithms. Journal of Computer and System Sciences 13, 335–379 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Brockenauer, R., Cornelsen, S.: Drawing Clusters and Hierarchies. In: Kaufmann, M., Wagner, D. (eds.) Drawing Graphs. LNCS, vol. 2025, ch. 8, pp. 193–227. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  3. 3.
    Chandramouli, M., Diwan, A.A.: Upward Numbering Testing for Triconnected Graphs. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 140–151. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  4. 4.
    Chiba, N., Nishizeki, T., Abe, S., Ozawa, T.: A Linear Algorithm for Embedding Planar Graphs Using PQ-Trees. Journal of Computer and System Sciences 30, 54–76 (1985)zbMATHCrossRefMathSciNetGoogle 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.
    Di Battista, G., Nardelli, E.: Hierarchies and Planarity Theory. IEEE Transactions on Systems, Man, and Cybernetics 18(6), 1035–1046 (1988)zbMATHCrossRefGoogle Scholar
  7. 7.
    Feng, Q.-W., Cohen, R.F., Eades, P.: Planarity for Clustered Graphs (Extended abstract). In: Spirakis, P.G. (ed.) ESA 1995. LNCS, vol. 979, pp. 213–226. Springer, Heidelberg (1995)Google Scholar
  8. 8.
    Forster, M.: Applying Crossing Reduction Strategies to Layered Compound Graphs. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 276–284. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    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
  10. 10.
    Healy, P., Kuusik, A.: The Vertex-Exchange Graph: A New Concept for Multi- Level Crossing Minimisation. In: Kratochvíl, J. (ed.) GD 1999. LNCS, vol. 1731, pp. 205–216. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. 11.
    Heath, L.S., Pemmaraju, S.V.: Recognizing Leveled-Planar Dags in Linear Time. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 300–311. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  12. 12.
    Heath, L.S., Pemmaraju, S.V.: Stack and Queue Layouts of Directed Acyclic Graphs: Part II. SIAM Journal on Computing 28(5), 1588–1626 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Jünger, M., Leipert, S.: Level Planar Embedding in Linear Time. In: Kratochvíl, J. (ed.) GD 1999. LNCS, vol. 1731, pp. 72–81. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  14. 14.
    Jünger, M., Leipert, S.: Level Planar Embedding in Linear Time. Journal of Graph Algorithms and Applications 6(1), 67–113 (2002)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Jünger, M., Leipert, S., Mutzel, P.: Level Planarity Testing in Linear Time. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 224–237. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  16. 16.
    Leipert, S.: Level Planarity Testing and Embedding in Linear Time. Dissertation, Mathematisch-Naturwissenschaftliche Fakultät der Universität zu Köln (1998)Google Scholar
  17. 17.
    Sander, G.: Layout of Compound Directed Graphs. Technical Report A/03/96, Universität Saarbrücken (1996)Google Scholar
  18. 18.
    Sander, G.: Visualisierungstechniken für den Compilerbau. PhD thesis, Universität Saarbrücken (1996)Google Scholar
  19. 19.
    Sander, G.: Graph Layout for Applications in Compiler Construction. Theoretical Computer Science 217, 175–214 (1999)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Michael Forster
    • 1
  • Christian Bachmaier
    • 1
  1. 1.University of PassauPassauGermany

Personalised recommendations