Advertisement

Clique-Width and Well-Quasi-Ordering of Triangle-Free Graph Classes

  • Konrad K. Dabrowski
  • Vadim V. Lozin
  • Daniël Paulusma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10520)

Abstract

Daligault, Rao and Thomassé asked whether every hereditary graph class that is well-quasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev (WG 2015) gave a negative answer to this question, but their counterexample is a class that can only be characterised by infinitely many forbidden induced subgraphs. This raises the issue of whether their question has a positive answer for finitely defined hereditary graph classes. Apart from two stubborn cases, this has been confirmed when at most two induced subgraphs \(H_1,H_2\) are forbidden. We confirm it for one of the two stubborn cases, namely for the case \((H_1,H_2)=(\text {triangle},P_2+P_4)\) by proving that the class of \((\text {triangle},P_2+P_4)\)-free graphs has bounded clique-width and is well-quasi-ordered. Our technique is based on a special decomposition of 3-partite graphs. We also use this technique to completely determine which classes of \((\text {triangle},H)\)-free graphs are well-quasi-ordered.

References

  1. 1.
    Atminas, A., Lozin, V.V.: Labelled induced subgraphs and well-quasi-ordering. Order 32(3), 313–328 (2015)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Blanché, A., Dabrowski, K.K., Johnson, M., Lozin, V.V., Paulusma, D., Zamaraev, V.: Clique-width for graph classes closed under complementation. In: Proceeding of MFCS 2017. LIPIcs, vol. 83, pp. 73:1–73:14 (2017)Google Scholar
  3. 3.
    Courcelle, B.: Clique-width and edge contraction. Inf. Process. Lett. 114(1–2), 42–44 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Appl. Math. 101(1–3), 77–114 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Dabrowski, K.K., Dross, F., Paulusma, D.: Colouring diamond-free graphs. J. Comput. Syst. Sci. (in press)Google Scholar
  7. 7.
    Dabrowski, K.K., Lozin, V.V., Paulusma, D.: Well-quasi-ordering versus clique-width: New results on bigenic classes. Order (in press)Google Scholar
  8. 8.
    Dabrowski, K.K., Paulusma, D.: Classifying the clique-width of \({H}\)-free bipartite graphs. Discret. Appl. Math. 200, 43–51 (2016)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Dabrowski, K.K., Paulusma, D.: Clique-width of graph classes defined by two forbidden induced subgraphs. Comput. J. 59(5), 650–666 (2016)CrossRefGoogle Scholar
  10. 10.
    Daligault, J., Rao, M., Thomassé, S.: Well-quasi-order of relabel functions. Order 27(3), 301–315 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Damaschke, P.: Induced subgraphs and well-quasi-ordering. J. Graph Theory 14(4), 427–435 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Ding, G.: Subgraphs and well-quasi-ordering. J. Graph Theory 16(5), 489–502 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Espelage, W., Gurski, F., Wanke, E.: How to solve NP-hard graph problems on clique-width bounded graphs in polynomial time. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 117–128. Springer, Heidelberg (2001). doi: 10.1007/3-540-45477-2_12 CrossRefGoogle Scholar
  14. 14.
    Higman, G.: Ordering by divisibility in abstract algebras. Proc. London Math. Soc. s3–2(1), 326–336 (1952)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Kamiński, M., Lozin, V.V., Milanič, M.: Recent developments on graphs of bounded clique-width. Discret. Appl. Math. 157(12), 2747–2761 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Kobler, D., Rotics, U.: Edge dominating set and colorings on graphs with fixed clique-width. Discrete Appl. Math. 126(2–3), 197–221 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Korpelainen, N., Lozin, V.V.: Bipartite induced subgraphs and well-quasi-ordering. J. Graph Theory 67(3), 235–249 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Korpelainen, N., Lozin, V.V.: Two forbidden induced subgraphs and well-quasi-ordering. Discret. Math. 311(16), 1813–1822 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Lozin, V.V., Rautenbach, D.: On the band-, tree-, and clique-width of graphs with bounded vertex degree. SIAM J. Discret Math. 18(1), 195–206 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Lozin, V.V., Razgon, I., Zamaraev, V.: Well-quasi-ordering does not imply bounded clique-width. In: Mayr, E.W. (ed.) WG 2015. LNCS, vol. 9224, pp. 351–359. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-53174-7_25 CrossRefGoogle Scholar
  21. 21.
    Oum, S.-I., Seymour, P.D.: Approximating clique-width and branch-width. J. Comb. Theory, Ser. B 96(4), 514–528 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Rao, M.: MSOL partitioning problems on graphs of bounded treewidth and clique-width. Theor. Comput. Sci. 377(1–3), 260–267 (2007)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Konrad K. Dabrowski
    • 1
  • Vadim V. Lozin
    • 2
  • Daniël Paulusma
    • 1
  1. 1.Department of Computer ScienceDurham UniversityDurhamUK
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK

Personalised recommendations