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Clique-Width: Harnessing the Power of Atoms

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Graph-Theoretic Concepts in Computer Science (WG 2020)

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Abstract

Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class \(\mathcal{G}\) if they are so on the atoms (graphs with no clique cut-set) of \(\mathcal{G}\). Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph G is H-free if H is not an induced subgraph of G, and it is \((H_1,H_2)\)-free if it is both \(H_1\)-free and \(H_2\)-free. A class of H-free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for \((H_1,H_2)\)-free graphs, as evidenced by one known example. We prove the existence of another such pair \((H_1,H_2)\) and classify the boundedness of clique-width on \((H_1,H_2)\)-free atoms for all but 18 cases.

The research in this paper received support from the Leverhulme Trust (RPG-2016-258). Masařík and Novotná were supported by Charles University student grants (SVV-2017-260452 and GAUK 1277018) and GAČR project (17-09142S). The last author was supported by Polish National Science Centre grant no. 2018/31/D/ST6/00062.

A preprint of the full version of this paper is available from arXiv  [29].

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Notes

  1. 1.

    See Sect. 2 for a definition of clique-width and other terminology used in Sect. 1.

References

  1. Alekseev, V.E.: On easy and hard hereditary classes of graphs with respect to the independent set problem. Discrete Appl. Math. 132(1–3), 17–26 (2003). https://doi.org/10.1016/S0166-218X(03)00387-1

    Article  MathSciNet  MATH  Google Scholar 

  2. Belmonte, R., Vatshelle, M.: Graph classes with structured neighborhoods and algorithmic applications. Theor. Comput. Sci. 511, 54–65 (2013). https://doi.org/10.1016/j.tcs.2013.01.011

    Article  MathSciNet  MATH  Google Scholar 

  3. Blanché, A., Dabrowski, K.K., Johnson, M., Lozin, V.V., Paulusma, D., Zamaraev, V.: Clique-width for graph classes closed under complementation. SIAM J. Discrete Math. 34(2), 1107–1147 (2020). https://doi.org/10.1137/18M1235016

    Article  MathSciNet  MATH  Google Scholar 

  4. Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996). https://doi.org/10.1137/S0097539793251219

    Article  MathSciNet  MATH  Google Scholar 

  5. Bodlaender, H.L., Brettell, N., Johnson, M., Paesani, G., Paulusma, D., van Leeuwen, E.J.: Steiner trees for hereditary graph classes. Proc. LATIN 2020, LNCS (2020, to appear)

    Google Scholar 

  6. 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). https://doi.org/10.1007/3-540-36136-7_5

    Chapter  Google Scholar 

  7. Brandstädt, A., Dabrowski, K.K., Huang, S., Paulusma, D.: Bounding the clique-width of \(H\)-free split graphs. Discrete Appl. Math. 211, 30–39 (2016). https://doi.org/10.1016/j.dam.2016.04.003

    Article  MathSciNet  MATH  Google Scholar 

  8. Brandstädt, A., Dabrowski, K.K., Huang, S., Paulusma, D.: Bounding the clique-width of \(H\)-free chordal graphs. J. Graph Theory 86(1), 42–77 (2017). https://doi.org/10.1002/jgt.22111

    Article  MathSciNet  MATH  Google Scholar 

  9. Brandstädt, A., Hoàng, C.T.: On clique separators, nearly chordal graphs, and the maximum weight stable set problem. Theor. Comput. Sci. 389(1–2), 295–306 (2007). https://doi.org/10.1016/j.tcs.2007.09.031

    Article  MathSciNet  MATH  Google Scholar 

  10. Brandstädt, A., Klembt, T., Mahfud, S.: \(P_6\)- and triangle-free graphs revisited: structure and bounded clique-width. Discrete Math. Theor. Comput. Sci. 8(1), 173–188 (2006). https://dmtcs.episciences.org/372

  11. 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). https://doi.org/10.1142/S0129054104002364

    Article  MathSciNet  MATH  Google Scholar 

  12. Brandstädt, A., Le, H.O., Mosca, R.: Chordal co-gem-free and (\(P_5\),gem)-free graphs have bounded clique-width. Discrete Appl. Math. 145(2), 232–241 (2005). https://doi.org/10.1016/j.dam.2004.01.014

    Article  MathSciNet  MATH  Google Scholar 

  13. 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. Inf. Process. Lett. 84(5), 251–259 (2002). https://doi.org/10.1016/S0020-0190(02)00291-0

    Article  MathSciNet  MATH  Google Scholar 

  14. Brandstädt, A., Mosca, R.: On distance-3 matchings and induced matchings. Discrete Appl. Math. 159(7), 509–520 (2011). https://doi.org/10.1016/j.dam.2010.05.022

    Article  MathSciNet  MATH  Google Scholar 

  15. Brettell, N., Horsfield, J., Munaro, A., Paesani, G., Paulusma, D.: Bounding the mim-width of hereditary graph classes. In: Cao, Y., Pilipczuk, M. (eds.) IPEC 2020. LIPIcs vol. 180, pp. 6:1–6:19 (2020). https://doi.org/10.4230/LIPIcs.IPEC.2020.6

  16. Bui-Xuan, B., Telle, J.A., Vatshelle, M.: Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems. Theor. Comput. Sci. 511, 66–76 (2013). https://doi.org/10.1016/j.tcs.2013.01.009

    Article  MathSciNet  MATH  Google Scholar 

  17. Cameron, K., da Silva, M.V.G., Huang, S., Vušković, K.: Structure and algorithms for (cap, even hole)-free graphs. Discrete Math. 341(2), 463–473 (2018). https://doi.org/10.1016/j.disc.2017.09.013

    Article  MathSciNet  MATH  Google Scholar 

  18. Chudnovsky, M., Seymour, P.D.: The structure of claw-free graphs. London Math. Soc. Lecture Note Series 327, 153–171 (2005). https://doi.org/10.1017/CBO9780511734885.008

    Article  MathSciNet  MATH  Google Scholar 

  19. Corneil, D.G., Rotics, U.: On the relationship between clique-width and treewidth. SIAM J. Comput. 34, 825–847 (2005). https://doi.org/10.1137/S0097539701385351

    Article  MathSciNet  MATH  Google Scholar 

  20. Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990). https://doi.org/10.1016/0890-5401(90)90043-H

  21. Courcelle, B., Engelfriet, J.: Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach, Encyclopedia of Mathematics and its Applications, vol. 138. Cambridge University Press (2012). https://doi.org/10.1017/CBO9780511977619

  22. 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). https://doi.org/10.1007/s002249910009

    Article  MathSciNet  MATH  Google Scholar 

  23. Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Appl. Math. 101(1–3), 77–114 (2000). https://doi.org/10.1016/S0166-218X(99)00184-5

    Article  MathSciNet  MATH  Google Scholar 

  24. Dabrowski, K.K., Dross, F., Paulusma, D.: Colouring diamond-free graphs. J. Comput. Syst. Sci. 89, 410–431 (2017). https://doi.org/10.1016/j.jcss.2017.06.005

    Article  MathSciNet  MATH  Google Scholar 

  25. Dabrowski, K.K., Huang, S., Paulusma, D.: Bounding clique-width via perfect graphs. J. Comput. Syst. Sci. 104, 202–215 (2019). https://doi.org/10.1016/j.jcss.2016.06.007

    Article  MathSciNet  MATH  Google Scholar 

  26. Dabrowski, K.K., Johnson, M., Paulusma, D.: Clique-width for hereditary graph classes. London Math. Soc. Lecture Note Series 456, 1–56 (2019). https://doi.org/10.1017/9781108649094.002

    Article  MathSciNet  Google Scholar 

  27. Dabrowski, K.K., Lozin, V.V., Paulusma, D.: Clique-width and well-quasi-ordering of triangle-free graph classes. J. Comput. Syst. Sci. 108, 64–91 (2020). https://doi.org/10.1016/j.jcss.2019.09.001

    Article  MathSciNet  MATH  Google Scholar 

  28. Dabrowski, K.K., Lozin, V.V., Raman, R., Ries, B.: Colouring vertices of triangle-free graphs without forests. Discrete Math. 312(7), 1372–1385 (2012). https://doi.org/10.1016/j.disc.2011.12.012

    Article  MathSciNet  MATH  Google Scholar 

  29. Dabrowski, K.K., Masařík, T., Novotná, J., Paulusma, D., Rzążewski, P.: Clique-width: Harnessing the power of atoms. CoRR abs/2006.03578 (2020). https://arxiv.org/abs/2006.03578

  30. Dabrowski, K.K., Paulusma, D.: Classifying the clique-width of \({H}\)-free bipartite graphs. Discrete Appl. Math. 200, 43–51 (2016). https://doi.org/10.1016/j.dam.2015.06.030

    Article  MathSciNet  MATH  Google Scholar 

  31. Dabrowski, K.K., Paulusma, D.: Clique-width of graph classes defined by two forbidden induced subgraphs. Comput. J. 59(5), 650–666 (2016). https://doi.org/10.1093/comjnl/bxv096

    Article  MathSciNet  Google Scholar 

  32. 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). https://doi.org/10.1007/3-540-45477-2_12

    Chapter  MATH  Google Scholar 

  33. Foley, A.M., Fraser, D.J., Hoàng, C.T., Holmes, K., LaMantia, T.P.: The intersection of two vertex coloring problems. Graphs Comb. 36(1), 125–138 (2020). https://doi.org/10.1007/s00373-019-02123-1

    Article  MathSciNet  MATH  Google Scholar 

  34. Fraser, D.J., Hamel, A.M., Hoàng, C.T., Holmes, K., LaMantia, T.P.: Characterizations of \((4K_1, C_4, C_5)\)-free graphs. Discrete Appl. Math. 231, 166–174 (2017). https://doi.org/10.1016/j.dam.2016.08.016

    Article  MathSciNet  MATH  Google Scholar 

  35. Gaspers, S., Huang, S., Paulusma, D.: Colouring square-free graphs without long induced paths. J. Comput. Syst. Sci. 106, 60–79 (2019). https://doi.org/10.1016/j.jcss.2019.06.002

    Article  MathSciNet  MATH  Google Scholar 

  36. Gerber, M.U., Kobler, D.: Algorithms for vertex-partitioning problems on graphs with fixed clique-width. Theor. Comput. Sci. 299(1), 719–734 (2003). https://doi.org/10.1016/S0304-3975(02)00725-9

    Article  MathSciNet  MATH  Google Scholar 

  37. Golovach, P.A., Johnson, M., Paulusma, D., Song, J.: A survey on the computational complexity of colouring graphs with forbidden subgraphs. J. Graph Theory 84(4), 331–363 (2017). https://doi.org/10.1002/jgt.22028

    Article  MathSciNet  MATH  Google Scholar 

  38. Grötschel, M., Lovász, L., Schrijver, A.: Polynomial algorithms for perfect graphs. Ann. Discrete Math. 21, 325–356 (1984). https://doi.org/10.1016/S0304-0208(08)72943-8

    Article  MathSciNet  MATH  Google Scholar 

  39. Gurski, F.: The behavior of clique-width under graph operations and graph transformations. Theory Comput. Syst. 60(2), 346–376 (2017). https://doi.org/10.1007/s00224-016-9685-1

    Article  MathSciNet  MATH  Google Scholar 

  40. Hermelin, D., Mnich, M., van Leeuwen, E.J., Woeginger, G.J.: Domination when the stars are out. ACM Trans. Algorithms 15(2), 25:1–25:90 (2019). https://doi.org/10.1145/3301445

  41. Hliněný, P., Oum, S., Seese, D., Gottlob, G.: Width parameters beyond tree-width and their applications. Comput. J. 51(3), 326–362 (2008). https://doi.org/10.1093/comjnl/bxm052

    Article  Google Scholar 

  42. Hoàng, C.T., Lazzarato, D.A.: Polynomial-time algorithms for minimum weighted colorings of \((P_5,\overline{P_5})\)-free graphs and similar graph classes. Discrete Appl. Math. 186, 106–111 (2015). https://doi.org/10.1016/j.dam.2015.01.022

    Article  MathSciNet  MATH  Google Scholar 

  43. Huang, S., Karthick, T.: On graphs with no induced five-vertex path or paraglider. CoRR abs/1903.11268 (2019). https://arxiv.org/abs/1903.11268

  44. Jansen, K., Scheffler, P.: Generalized coloring for tree-like graphs. Discrete Appl. Math. 75(2), 135–155 (1997). https://doi.org/10.1016/S0166-218X(96)00085-6

    Article  MathSciNet  MATH  Google Scholar 

  45. Kamiński, M., Lozin, V.V., Milanič, M.: Recent developments on graphs of bounded clique-width. Discrete Appl. Math. 157(12), 2747–2761 (2009). https://doi.org/10.1016/j.dam.2008.08.022

    Article  MathSciNet  MATH  Google Scholar 

  46. Kobler, D., Rotics, U.: Edge dominating set and colorings on graphs with fixed clique-width. Discrete Appl. Math. 126(2–3), 197–221 (2003). https://doi.org/10.1016/S0166-218X(02)00198-1

    Article  MathSciNet  MATH  Google Scholar 

  47. Lozin, V.V., Rautenbach, D.: On the band-, tree-, and clique-width of graphs with bounded vertex degree. SIAM J. Discrete Math. 18(1), 195–206 (2004). https://doi.org/10.1137/S0895480102419755

    Article  MathSciNet  MATH  Google Scholar 

  48. Makowsky, J.A., Rotics, U.: On the clique-width of graphs with few \(P_4\)’s. Int. J. Found. Comput. Sci. 10(03), 329–348 (1999). https://doi.org/10.1142/S0129054199000241

    Article  MATH  Google Scholar 

  49. Malyshev, D.S., Lobanova, O.O.: Two complexity results for the vertex coloring problem. Discrete Appl. Math. 219, 158–166 (2017). https://doi.org/10.1016/j.dam.2016.10.025

    Article  MathSciNet  MATH  Google Scholar 

  50. Oum, S., Seymour, P.D.: Approximating clique-width and branch-width. J. Comb. Theory Ser. B 96(4), 514–528 (2006). https://doi.org/10.1016/j.jctb.2005.10.006

    Article  MathSciNet  MATH  Google Scholar 

  51. Rao, M.: MSOL partitioning problems on graphs of bounded treewidth and clique-width. Theor. Comput. Sci. 377(1–3), 260–267 (2007). https://doi.org/10.1016/j.tcs.2007.03.043

    Article  MathSciNet  MATH  Google Scholar 

  52. Sæther, S.H., Vatshelle, M.: Hardness of computing width parameters based on branch decompositions over the vertex set. Theor. Comput. Sci. 615, 120–125 (2016). https://doi.org/10.1016/j.tcs.2015.11.039

    Article  MathSciNet  MATH  Google Scholar 

  53. Tarjan, R.E.: Decomposition by clique separators. Discrete Math. 55(2), 221–232 (1985). https://doi.org/10.1016/0012-365X(85)90051-2

    Article  MathSciNet  MATH  Google Scholar 

  54. Vatshelle, M.: New Width Parameters of Graphs. Ph.D. thesis, University of Bergen (2012)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Durham University, Durham, UK

    Konrad K. Dabrowski & Daniël Paulusma

  2. Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Tomáš Masařík & Jana Novotná

  3. Institute of Informatics, University of Warsaw, Warsaw, Poland

    Tomáš Masařík, Jana Novotná & Paweł Rzążewski

  4. Department of Mathematics, Simon Fraser University, Burnaby, Canada

    Tomáš Masařík

  5. Faculty of Mathematics and Information Science, Warsaw University of Technology, Warsaw, Poland

    Paweł Rzążewski

Authors
  1. Konrad K. Dabrowski
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  2. Tomáš Masařík
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  3. Jana Novotná
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  4. Daniël Paulusma
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  5. Paweł Rzążewski
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Correspondence to Konrad K. Dabrowski .

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  1. University of Leeds, Leeds, UK

    Isolde Adler

  2. University of Leeds, Leeds, UK

    Haiko Müller

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Dabrowski, K.K., Masařík, T., Novotná, J., Paulusma, D., Rzążewski, P. (2020). Clique-Width: Harnessing the Power of Atoms. In: Adler, I., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2020. Lecture Notes in Computer Science(), vol 12301. Springer, Cham. https://doi.org/10.1007/978-3-030-60440-0_10

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