On Embedding a Cycle in a Plane Graph

(Extended Abstract)
  • Pier Francesco Cortese
  • Giuseppe Di Battista
  • Maurizio Patrignani
  • Maurizio Pizzonia
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3843)


Consider a planar drawing \({\it \Gamma}\) of a planar graph G such that the vertices are drawn as small circles and the edges are drawn as thin strips. Consider a cycle c of G. Is it possible to draw c as a non-intersecting closed curve inside \({\it \Gamma}\), following the circles that correspond in \({\it \Gamma}\) to the vertices of c and the strips that connect them? We show that this test can be done in polynomial time and study this problem in the framework of clustered planarity for highly non-connected clustered graphs.


  1. 1.
    Biedl, T.C.: Drawing planar partitions III: Two constrained embedding problems. Tech. Report RRR 13-98, RUTCOR Rutgen University (1998)Google Scholar
  2. 2.
    Biedl, T.C., Kaufmann, M., Mutzel, P.: Drawing planar partitions II: HH-Drawings. In: Hromkovič, J., Sýkora, O. (eds.) WG 1998. LNCS, vol. 1517, pp. 124–136. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  3. 3.
    Cornelsen, S., Wagner, D.: Completely connected clustered graphs. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 168–179. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    Cortese, P.F., Di Battista, G.: Clustered planarity. In: SCG 2005: Proceedings of the twenty-first annual symposium on Computational geometry, pp. 32–34. ACM Press, New York (2005)CrossRefGoogle Scholar
  5. 5.
    Cortese, P.F., Di Battista, G., 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
  6. 6.
    Dahlhaus, E.: 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
  7. 7.
    Di Battista, G., 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
  8. 8.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice Hall, Upper Saddle River (1999)MATHGoogle Scholar
  9. 9.
    Even, S.: Graph Algorithms. Computer Science Press, Potomac (1979)MATHGoogle Scholar
  10. 10.
    Feng, Q.W., Cohen, R.F., Eades, P.: How to draw a planar clustered graph. In: Li, M., Du, D.-Z. (eds.) COCOON 1995. LNCS, vol. 959, pp. 21–30. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  11. 11.
    Feng, Q.W., Cohen, R.F., Eades, P.: Planarity for clustered graphs. In: Spirakis, P.G. (ed.) ESA 1995. LNCS, vol. 979, pp. 213–226. Springer, Heidelberg (1995)Google Scholar
  12. 12.
    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
  13. 13.
    Lengauer, T.: Hierarchical planarity testing algorithms. J. ACM 36(3), 474–509 (1989)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Pier Francesco Cortese
    • 1
  • Giuseppe Di Battista
    • 1
  • Maurizio Patrignani
    • 1
  • Maurizio Pizzonia
    • 1
  1. 1.Università Roma Tre 

Personalised recommendations