Clique-Width of Graph Classes Defined by Two Forbidden Induced Subgraphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9079)


If a graph has no induced subgraph isomorphic to any graph in a finite family \(\{H_1,\ldots ,H_p\}\), it is said to be \((H_1,\ldots ,H_p)\)-free. The class of \(H\)-free graphs has bounded clique-width if and only if \(H\) is an induced subgraph of the 4-vertex path \(P_4\). We study the (un)boundedness of the clique-width of graph classes defined by two forbidden induced subgraphs \(H_1\) and \(H_2\). Prior to our study it was not known whether the number of open cases was finite. We provide a positive answer to this question. To reduce the number of open cases we determine new graph classes of bounded clique-width and new graph classes of unbounded clique-width. For obtaining the latter results we first present a new, generic construction for graph classes of unbounded clique-width. Our results settle the boundedness or unboundedness of the clique-width of the class of \((H_1,H_2)\)-free graphs
  1. (i)

    for all pairs \((H_1,H_2)\), both of which are connected, except two non-equivalent cases, and

  2. (ii)

    for all pairs \((H_1,H_2)\), at least one of which is not connected, except 11 non-equivalent cases.


We also consider classes characterized by forbidding a finite family of graphs \(\{H_1,\ldots ,H_p\}\) as subgraphs, minors and topological minors, respectively, and completely determine which of these classes have bounded clique-width. Finally, we show algorithmic consequences of our results for the graph colouring problem restricted to \((H_1,H_2)\)-free graphs.


Clique-width Forbidden induced subgraph Graph class 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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., Dabrowski, K.K., Huang, S., Paulusma, D.: Bounding the clique-width of \(H\)-free chordal graphs. CoRR, abs/1502.06948 (2015)Google Scholar
  4. 4.
    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
  5. 5.
    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
  6. 6.
    Brandstädt, A., Kratsch, D.: On the structure of (\(P_5\), gem)-free graphs. Discrete Applied Mathematics 145(2), 155–166 (2005)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    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
  8. 8.
    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
  9. 9.
    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
  10. 10.
    Brandstädt, A., Mosca, R.: On variations of \(P_4\)-sparse graphs. Discrete Applied Mathematics 129(2–3), 521–532 (2003)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Broersma, H., Golovach, P.A., Paulusma, D., Song, J.: Determining the chromatic number of triangle-free \(2P_3\)-free graphs in polynomial time. Theoretical Computer Science 423, 1–10 (2012)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Broersma, H., Golovach, P.A., Paulusma, D., Song, J.: Updating the complexity status of coloring graphs without a fixed induced linear forest. Theoretical Computer Science 414(1), 9–19 (2012)CrossRefMATHMathSciNetGoogle 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., Huang, S., Paulusma, D.: Bounding Clique-Width via Perfect Graphs. In: Dediu, A.-H., Formenti, E., Martín-Vide, C., Truthe, B. (eds.) LATA 2015. LNCS, vol. 8977, pp. 676–688. Springer, Heidelberg (2015). Full version: arXiv CoRR abs/1406.6298 CrossRefGoogle Scholar
  16. 16.
    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
  17. 17.
    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
  18. 18.
    de Ridder, H.N., et al. Information System on Graph Classes and their Inclusions, 2001–2013.
  19. 19.
    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) CrossRefGoogle Scholar
  20. 20.
    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
  21. 21.
    Golovach, P.A., Johnson, M., Paulusma, D., Song, J.: A survey on the computational complexity of colouring graphs with forbidden subgraphs. CoRR, abs/1407.1482 (2014)Google Scholar
  22. 22.
    Golovach, P.A., Paulusma, D.: List coloring in the absence of two subgraphs. Discrete Applied Mathematics 166, 123–130 (2014)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    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
  24. 24.
    Gurski, F.: Graph operations on clique-width bounded graphs. CoRR, abs/cs/0701185 (2007)Google Scholar
  25. 25.
    Hoàng, C.T., Lazzarato, D.A.: Polynomial-time algorithms for minimum weighted colorings of \((P_5,\overline{P_5})\)-free graphs and related graph classes. Discrete Applied Mathematics 186(0166–218X), 106–111 (2015). doi:
  26. 26.
    Huang, S., Johnson, M., Paulusma, D.: Narrowing the complexity gap for colouring \(({C}_s,{P}_t)\)-free graphs. In: Gu, Q., Hell, P., Yang, B. (eds.) AAIM 2014. LNCS, vol. 8546, pp. 162–173. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  27. 27.
    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
  28. 28.
    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
  29. 29.
    Král’, D., Kratochvíl, J., Tuza, Z., Woeginger, G.J.: Complexity of Coloring Graphs without Forbidden Induced Subgraphs. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 254–262. Springer, Heidelberg (2001) CrossRefGoogle Scholar
  30. 30.
    Kratsch, S., Schweitzer, P.: Graph Isomorphism for Graph Classes Characterized by Two Forbidden Induced Subgraphs. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds.) WG 2012. LNCS, vol. 7551, pp. 34–45. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  31. 31.
    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
  32. 32.
    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
  33. 33.
    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
  34. 34.
    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
  35. 35.
    Malyshev, D.S.: The coloring problem for classes with two small obstructions. Optimization Letters 8(8), 2261–2270 (2014)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Malyshev, D.S.: Two cases of polynomial-time solvability for the coloring problem. Journal of Combinatorial Optimization (in press)Google Scholar
  37. 37.
    Olariu, S.: Paw-free graphs. Information Processing Letters 28(1), 53–54 (1988)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Oum, S.-I.: Approximating rank-width and clique-width quickly. ACM Transactions on Algorithms 5(1), 10 (2008)CrossRefMathSciNetGoogle Scholar
  39. 39.
    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
  40. 40.
    Ramsey, F.P.: On a problem of formal logic. Proceedings of the London Mathematical Society s2–30(1), 264–286 (1930)Google Scholar
  41. 41.
    Randerath, B.: 3-colorability and forbidden subgraphs. I: Characterizing pairs. Discrete Mathematics 276(1–3), 313–325 (2004)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Randerath, B., Schiermeyer, I.: A note on Brooks’ theorem for triangle-free graphs. Australasian Journal of Combinatorics 26, 3–9 (2002)MATHMathSciNetGoogle Scholar
  43. 43.
    Rao, M.: MSOL partitioning problems on graphs of bounded treewidth and clique-width. Theoretical Computer Science 377(1–3), 260–267 (2007)CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Schindl, D.: Some new hereditary classes where graph coloring remains NP-hard. Discrete Mathematics 295(1–3), 197–202 (2005)CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Schweitzer, P.: Towards an isomorphism dichotomy for hereditary graph classes. In: Mayr, E.W., Ollinger, N. (eds.) 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). LIPIcs, vol. 30, pp. 689–702. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Dagstuhl (2015). doi:
  46. 46.
    Stein, W.A., et al.: Sage Mathematics Software (Version 5.9). The Sage Development Team (2013).

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.School of Engineering and Computing SciencesDurham University Science LaboratoriesDurhamUK

Personalised recommendations