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Classifying the Clique-Width of H-Free Bipartite Graphs

  • Konrad Kazimierz Dabrowski
  • Daniël Paulusma
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8591)

Abstract

Let G be a bipartite graph, and let H be a bipartite graph with a fixed bipartition (B H ,W H ). We consider three different, natural ways of forbidding H as an induced subgraph in G. First, G is H-free if it does not contain H as an induced subgraph. Second, G is strongly H-free if G is H-free or else has no bipartition (B G ,W G ) with B H  ⊆ B G and W H  ⊆ W G . Third, G is weakly H-free if G is H-free or else has at least one bipartition (B G ,W G ) with \(B_H\not\subseteq B_G\) or \(W_H\not\subseteq W_G\). Lozin and Volz characterized all bipartite graphs H for which the class of strongly H-free bipartite graphs has bounded clique-width. We extend their result by giving complete classifications for the other two variants of H-freeness.

Keywords

Bipartite Graph Graph Class Free Graph Discrete Apply Mathematic Graph Operation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Boliac, R., Lozin, V.: On the clique-width of graphs in hereditary classes. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 44–54. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Bonomo, F., Grippo, L.N., Milanič, M., Safe, M.D.: Graphs of power-bounded clique-width. arXiv, abs/1402.2135 (2014)Google Scholar
  3. 3.
    Brandstädt, A., Engelfriet, J., Le, H.-O., Lozin, V.: Clique-width for 4-vertex forbidden subgraphs. Theory of Computing Systems 39(4), 561–590 (2006)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Brandstädt, A., Klembt, T., Mahfud, S.: P 6- and triangle-free graphs revisited: structure and bounded clique-width. Discrete Mathematics and Theoretical Computer Science 8(1), 173–188 (2006)MathSciNetGoogle Scholar
  5. 5.
    Brandstädt, A., Kratsch, D.: On the structure of (P 5,gem)-free graphs. Discrete Applied Mathematics 145(2), 155–166 (2005)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Brandstädt, A., Le, H.-O., Mosca, R.: Gem- and co-gem-free graphs have bounded clique-width. International Journal of Foundations of Computer Science 15(1), 163–185 (2004)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Brandstädt, A., Le, H.-O., Mosca, R.: Chordal co-gem-free and (P 5,gem)-free graphs have bounded clique-width. Discrete Applied Mathematics 145(2), 232–241 (2005)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Brandstädt, A., Mahfud, S.: Maximum weight stable set on graphs without claw and co-claw (and similar graph classes) can be solved in linear time. Information Processing Letters 84(5), 251–259 (2002)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Brandstädt, A., Mosca, R.: On variations of P 4-sparse graphs. Discrete Applied Mathematics 129(2-3), 521–532 (2003)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory of Computing Systems 33(2), 125–150 (2000)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Dabrowski, K.K., Golovach, P.A., Paulusma, D.: Colouring of graphs with Ramsey-type forbidden subgraphs. Theoretical Computer Science 522, 34–43 (2013)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Dabrowski, K.K., Paulusma, D.: Clique-width of graph classes defined by two forbidden induced subgraphs. coRR, abs/1405.7092 (2014)Google Scholar
  13. 13.
    Gurski, F.: Graph operations on clique-width bounded graphs. CoRR, abs/cs/0701185 (2007)Google Scholar
  14. 14.
    Kamiński, M., Lozin, V., Milanič, M.: Recent developments on graphs of bounded clique-width. Discrete Applied Mathematics 157(12), 2747–2761 (2009)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Kobler, D., Rotics, U.: Edge dominating set and colorings on graphs with fixed clique-width. Discrete Applied Mathematics 126(2-3), 197–221 (2003)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Lozin, V., Rautenbach, D.: On the band-, tree-, and clique-width of graphs with bounded vertex degree. SIAM Journal on Discrete Mathematics 18(1), 195–206 (2004)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Lozin, V., Rautenbach, D.: The tree- and clique-width of bipartite graphs in special classes. Australasian Journal of Combinatorics 34, 57–67 (2006)MATHMathSciNetGoogle Scholar
  18. 18.
    Lozin, V., Volz, J.: The clique-width of bipartite graphs in monogenic classes. International Journal of Foundations of Computer Science 19(02), 477–494 (2008)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Makowsky, J., Rotics, U.: On the clique-width of graphs with few P 4’s. International Journal of Foundations of Computer Science 10(3), 329–348 (1999)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Oum, S.-I.: Approximating rank-width and clique-width quickly. ACM Trans. Algorithms 5(1), 10:1–10:20 (2008)Google Scholar
  21. 21.
    Rao, M.: MSOL partitioning problems on graphs of bounded treewidth and clique-width. Theoretical Computer Science 377(1-3), 260–267 (2007)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Konrad Kazimierz Dabrowski
    • 1
  • Daniël Paulusma
    • 1
  1. 1.School of Engineering and Computing SciencesDurham University, Science LaboratoriesDurhamUnited Kingdom

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