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)


Let G be a bipartite graph, and let H be a bipartite graph with a fixed bipartition (BH,WH). 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 (BG,WG) with BH ⊆ BG and WH ⊆ WG. Third, G is weakly H-free if G is H-free or else has at least one bipartition (BG,WG) 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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

Personalised recommendations