Order

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Well-Quasi-Ordering versus Clique-Width: New Results on Bigenic Classes

  • Konrad K. Dabrowski
  • Vadim V. Lozin
  • Daniël Paulusma
Open Access
Article

Abstract

Daligault, Rao and Thomassé asked whether a hereditary class of graphs well-quasi-ordered by the induced subgraph relation has bounded clique-width. Lozin, Razgon and Zamaraev recently showed that this is not true for classes defined by infinitely many forbidden induced subgraphs. However, in the case of finitely many forbidden induced subgraphs the question remains open and we conjecture that in this case the answer is positive. The conjecture is known to hold for classes of graphs defined by a single forbidden induced subgraph H, as such graphs are well-quasi-ordered and are of bounded clique-width if and only if H is an induced subgraph of P4. For bigenic classes of graphs, i.e. ones defined by two forbidden induced subgraphs, there are several open cases in both classifications. In the present paper we obtain a number of new results on well-quasi-orderability of bigenic classes, each of which supports the conjecture.

Keywords

Well-quasi-order Induced subgraph Hereditary graph class Bigenic class 

References

  1. 1.
    Atminas, A., Lozin, V.V.: Labelled induced subgraphs and well-quasi-ordering. Order 32(3), 313–328 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brandstädt, A., Kratsch, D.: On the structure of (p 5,gem)-free graphs. Discret. Appl. Math. 145(2), 155–166 (2005)CrossRefMATHGoogle Scholar
  4. 4.
    Brandstädt, A., Le, H.-O., Mosca, R.: Gem- and co-gem-free graphs have bounded clique-width. Int. J. Found. Comput. Sci. 15(1), 163–185 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brandstädt, A., Le, H.-O., Mosca, R.: Chordal co-gem-free and (p 5,gem)-free graphs have bounded clique-width. Discret. Appl. Math. 145(2), 232–241 (2005)CrossRefMATHGoogle Scholar
  6. 6.
    Courcelle, B.: The monadic second-order logic of graphs III:, tree-decompositions, minor and complexity issues. Informatique Théorique et Applications 26(3), 257–286 (1992)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Courcelle, B.: Clique-width and edge contraction. Inf. Process. Lett. 114(1–2), 42–44 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discret. Appl. Math. 101(1–3), 77–114 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dabrowski, K.K., Dross, F., Paulusma, D.: Colouring diamond-free graphs. Proc. SWAT, 2016 LIPIcs 53, 16:1–16:14 (2016)MathSciNetGoogle Scholar
  11. 11.
    Dabrowski, K.K., Huang, S., Paulusma, D.: Bounding clique-width via perfect graphs. Journal of Computer and System Sciences, (in press)Google Scholar
  12. 12.
    Dabrowski, K.K., Lozin, V.V., Paulusma, D.: Well-quasi-ordering versus clique-width: New results on bigenic classes. Proc. IWOCA 2016, LNCS 9843, 253–265 (2016)Google Scholar
  13. 13.
    Dabrowski, K.K., Lozin, V.V., Paulusma, D.: Clique-Width and Well-Quasi Ordering of Triangle-Free Graph Classes. Proceedings WG 2017 LNCS, (to appear)Google Scholar
  14. 14.
    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
  15. 15.
    Daligault, J., Rao, M., Thomassé, S.: Well-quasi-order of relabel functions. Order 27(3), 301–315 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Damaschke, P.: Induced subgraphs and well-quasi-ordering. Journal of Graph Theory 14(4), 427–435 (1990)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Espelage, W., Gurski, F., Wanke, E.: How to solve NP-hard graph problems on clique-width bounded graphs in polynomial time. Proc. WG 2001, LNCS 2204, 117–128 (2001)MathSciNetMATHGoogle Scholar
  18. 18.
    Finkel, A., Schnoebelen, P.: Well-structured transition systems everywhere!. Theor. Comput. Sci. 256(1–2), 63–92 (2001)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Higman, G.: Ordering by divisibility in abstract algebras. Proc. Lond. Math. Soc. s3–2(1), 326–336 (1952)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kobler, D., Rotics, U.: Edge dominating set and colorings on graphs with fixed clique-width. Discret. Appl. Math. 126(2–3), 197–221 (2003)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Korpelainen, N., Lozin, V.V.: Two forbidden induced subgraphs and well-quasi-ordering. Discret. Math. 311(16), 1813–1822 (2011)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kruskal, J.B.: The theory of well-quasi-ordering: A frequently discovered concept. Journal of Combinatorial Theory Series A 13(3), 297–305 (1972)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lozin, V.V., Razgon, I., Zamaraev, V.: Well-quasi-ordering does not imply bounded clique-width. Proc. WG 2015, LNCS 9224, 351–359 (2016)MathSciNetMATHGoogle Scholar
  24. 24.
    Oum, S.-I., Seymour, P.D.: Approximating clique-width and branch-width. Journal of Combinatorial Theory Series B 96(4), 514–528 (2006)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Rao, M.: MSOL partitioning problems on graphs of bounded treewidth and clique-width. Theor. Comput. Sci. 377(1–3), 260–267 (2007)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Robertson, N., Seymour, P.D.: Graph minors. IV. Tree-width and well-quasi-ordering. Journal of Combinatorial Theory Series B 48(2), 227–254 (1990)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Robertson, N., Seymour, P.D.: Graph minors. XX. Wagner’s conjecture. Journal of Combinatorial Theory Series B 92(2), 325–357 (2004)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Konrad K. Dabrowski
    • 1
  • Vadim V. Lozin
    • 2
  • Daniël Paulusma
    • 1
  1. 1.School of Engineering and Computing SciencesDurham UniversityDurhamUK
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK

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