Non-planar Core Reduction of Graphs

  • Carsten Gutwenger
  • Markus Chimani
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3843)

Abstract

We present a reduction method that reduces a graph to a smaller core graph which behaves invariant with respect to planarity measures like crossing number, skewness, and thickness. The core reduction is based on the decomposition of a graph into its triconnected components and can be computed in linear time. It has applications in heuristic and exact optimization algorithms for the planarity measures mentioned above. Experimental results show that this strategy yields a reduction to 2/3 in average for a widely used benchmark set of graphs.

References

  1. 1.
    Chimani, M., Gutwenger, C.: On the minimum cut of planarizations. Technical report, University of Dortmund, Germany (2005)Google Scholar
  2. 2.
    Di Battista, G., Garg, A., Liotta, G., Tamassia, R., Tassinari, E., Vargiu, F.: An experimental comparison of four graph drawing algorithms. Comput. Geom. Theory Appl. 7, 303–325 (1997)MATHGoogle Scholar
  3. 3.
    Di Battista, G., Tamassia, R.: On-line maintanance of triconnected components with SPQR-trees. Algorithmica 15, 302–318 (1996)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Di Battista, G., Tamassia, R.: On-line planarity testing. SIAM J. Comput. 25(5), 956–997 (1996)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods 4(3), 312–316 (1983)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gutwenger, C., Mutzel, P.: A linear time implementation of SPQR trees. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 77–90. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Gutwenger, C., Mutzel, P.: An experimental study of crossing minimization heuristics. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 13–24. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. 8.
    Gutwenger, C., Mutzel, P., Weiskircher, R.: Inserting an edge into a planar graph. Algorithmica 41(4), 289–308 (2005)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hopcroft, J., Tarjan, R.E.: Efficient planarity testing. J. ACM 21(4), 549–568 (1974)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kuratowski, C.: Sur le problème des courbes gauches en topologie. Fundamenta Mathematicae 15, 271–283 (1930)MATHGoogle Scholar
  11. 11.
    Liebers, A.: Planarizing graphs — A survey and annotated bibliography. J. Graph Algorithms and Applications 5(1), 1–74 (2001)MathSciNetGoogle Scholar
  12. 12.
    Liu, P.C., Geldmacher, R.C.: On the deletion of nonplanar edges of a graph. Congr. Numerantium 24, 727–738 (1979)MathSciNetGoogle Scholar
  13. 13.
    Mansfield, A.: Determining the thickness of graphs is NP-hard. Math. Proc. Camb. Philos. Soc. 93, 9–23 (1983)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Mutzel, P., Odenthal, T., Scharbrodt, M.: The thickness of graphs: A survey. Graphs and Combinatorics 14(1), 59–73 (1998)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Širáň, J.: Additivity of the crossing number of graphs with connectivity 2. Periodica Mathematica Hungarica 15(4), 301–305 (1984)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Carsten Gutwenger
    • 1
  • Markus Chimani
    • 1
  1. 1.University of DortmundDortmundGermany

Personalised recommendations