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Computing Maximum C-Planar Subgraphs

  • Markus Chimani
  • Carsten Gutwenger
  • Mathias Jansen
  • Karsten Klein
  • Petra Mutzel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5417)

Abstract

Deciding c-planarity for a given clustered graph C = (G,T) is one of the most challenging problems in current graph drawing research. Though it is yet unknown if this problem is solvable in polynomial time, latest research focused on algorithmic approaches for special classes of clustered graphs. In this paper, we introduce an approach to solve the general problem using integer linear programming (ILP) techniques. We give an ILP formulation that also includes the natural generalization of c-planarity testing—the maximum c-planar subgraph problem—and solve this ILP with a branch-and-cut algorithm. Our computational results show that this approach is already successful for many clustered graphs of small to medium sizes and thus can be the foundation of a practically efficient algorithm that integrates further sophisticated ILP techniques.

References

  1. 1.
    Chimani, M., Mutzel, P., Schmidt, J.M.: Efficient extraction of multiple kuratowski subdivisions. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 159–170. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Cornelsen, S., Wagner, D.: Completely connected clustered graphs. J. Discrete Algorithms 4(2), 313–323 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    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
  4. 4.
    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
  5. 5.
    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
  6. 6.
    Di Battista, G., Frati, F.: Efficient c-planarity testing for embedded flat clustered graphs with small faces. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 291–302. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    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(5-6), 303–325 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    Goodrich, M.T., Lueker, G.S., Sun, J.Z.: C-planarity of extrovert clustered graphs. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 211–222. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    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
  11. 11.
    Jelínková, E., Kára, J., Kratochvíl, J., Pergel, M., Suchý, O., Vyskocil, T.: Clustered planarity: Small clusters in eulerian graphs. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 303–314. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Jünger, M., Mutzel, P.: Maximum planar subgraphs and nice embeddings: Practical layout tools. Algorithmica 16(1), 33–59 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Jünger, M., Thienel, S.: The ABACUS system for branch-and-cut-and-price algorithms in integer programming and combinatorial optimization. Software: Practice and Experience 30, 1325–1352 (2000)zbMATHGoogle Scholar
  14. 14.
    Kuratowski, K.: Sur le problème des courbes gauches en topologie. Fundamenta Mathematicae 15, 271–283 (1930)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Markus Chimani
    • 1
  • Carsten Gutwenger
    • 1
  • Mathias Jansen
    • 1
  • Karsten Klein
    • 1
  • Petra Mutzel
    • 1
  1. 1.Technische Universität DortmundGermany

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