Bounding Clique-Width via Perfect Graphs

  • Konrad Kazimierz Dabrowski
  • Shenwei Huang
  • Daniël Paulusma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8977)

Abstract

Given two graphs \(H_1\) and \(H_2\), a graph \(G\) is \((H_1,H_2)\)-free if it contains no subgraph isomorphic to \(H_1\) or \(H_2\). We continue a recent study into the clique-width of \((H_1,H_2)\)-free graphs and present three new classes of \((H_1,H_2)\)-free graphs that have bounded clique-width. We also show the implications of our results for the computational complexity of the Colouring problem restricted to \((H_1,H_2)\)-free graphs. The three new graph classes have in common that one of their two forbidden induced subgraphs is the diamond (the graph obtained from a clique on four vertices by deleting one edge). To prove boundedness of their clique-width we develop a technique based on bounding clique covering number in combination with reduction to subclasses of perfect graphs.

Keywords

Clique-width Forbidden induced subgraphs Graph class 

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References

  1. 1.
    Boliac, R., Lozin, V.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.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.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Annals of Mathematics 164, 51–229 (2006)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Corneil, D.G., Rotics, U.: On the relationship between clique-width and treewidth. SIAM Journal on Computing 34, 825–847 (2005)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Courcelle, B.: Clique-width and edge contraction. Information Processing Letters 114(1–2), 42–44 (2014)CrossRefMathSciNetGoogle Scholar
  13. 13.
    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
  14. 14.
    Dabrowski, K.K., Golovach, P.A., Paulusma, D.: Colouring of graphs with Ramsey-type forbidden subgraphs. Theoretical Computer Science 522, 34–43 (2014)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Dabrowski, K.K., Lozin, V.V., Raman, R., Ries, B.: Colouring vertices of triangle-free graphs without forests. Discrete Mathematics 312(7), 1372–1385 (2012)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Dabrowski, K.K., Paulusma, D.: Classifying the clique-width of H-free bipartite graphs. In: Cai, Z., Zelikovsky, A., Bourgeois, A. (eds.) COCOON 2014. LNCS, vol. 8591, pp. 489–500. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  17. 17.
    Dabrowski, K.K., Paulusma, D.: Clique-width of graph classes defined by two forbidden induced subgraphs. CoRR abs/1405.7092 (2014)Google Scholar
  18. 18.
    Fellows, M.R., Rosamond, F.A., Rotics, U., Szeider, S.: Clique-width is NP-Complete. SIAM Journal on Discrete Mathematics 23(2), 909–939 (2009)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Golumbic, M.C., Rotics, U.: On the clique-width of some perfect graph classes. International Journal of Foundations of Computer Science 11(03), 423–443 (2000)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Gurski, F.: Graph operations on clique-width bounded graphs. CoRR abs/cs/0701185 (2007)Google Scholar
  21. 21.
    Johansson, Ö.: Clique-decomposition, NLC-decomposition, and modular decomposition - relationships and results for random graphs. Congressus Numerantium 132, 39–60 (1998)MATHMathSciNetGoogle Scholar
  22. 22.
    Kamiński, M., Lozin, V.V., Milanič, M.: Recent developments on graphs of bounded clique-width. Discrete Applied Mathematics 157(12), 2747–2761 (2009)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    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
  24. 24.
    Lozin, V.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
  25. 25.
    Lozin, V.V., Rautenbach, D.: The tree- and clique-width of bipartite graphs in special classes. Australasian Journal of Combinatorics 34, 57–67 (2006)MATHMathSciNetGoogle Scholar
  26. 26.
    Lozin, V.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
  27. 27.
    Makowsky, J.A., Rotics, U.: On the clique-width of graphs with few \(P_4\)’s. International Journal of Foundations of Computer Science 10(03), 329–348 (1999)CrossRefMathSciNetGoogle Scholar
  28. 28.
    Oum, S.I.: Approximating rank-width and clique-width quickly. ACM Transactions on Algorithms 5(1), 10 (2008)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Oum, S.I., Seymour, P.D.: Approximating clique-width and branch-width. Journal of Combinatorial Theory, Series B 96(4), 514–528 (2006)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    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 2015

Authors and Affiliations

  • Konrad Kazimierz Dabrowski
    • 1
  • Shenwei Huang
    • 2
  • Daniël Paulusma
    • 1
  1. 1.School of Engineering and Computing SciencesDurham UniversityDurhamUK
  2. 2.School of Computing ScienceSimon Fraser UniversityBurnabyCanada

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