Partitioning Graphs into Induced Subgraphs

  • Dušan KnopEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10168)


We study the Partition into \(H\) problem from the parametrized complexity point of view. In the Partition into \(H\) problem the task is to partition the vertices of a graph G into sets \(V_1,\dots ,V_r\) such that the graph H is isomorphic to the subgraph of G induced by each set \(V_i\) for \(i = 1,\dots ,r.\) The pattern graph H is fixed.

For the parametrization we consider three distinct structural parameters of the graph G – namely the tree-width, the neighborhood diversity, and the modular-width. For the parametrization by the neighborhood diversity we obtain an FPT algorithm for every graph H. For the parametrization by the tree-width we obtain an FPT algorithm for every connected graph H. Thus resolving an open question of Gajarský et al. from IPEC 2013 [9]. Finally, for the parametrization by the modular-width we derive an FPT algorithm for every prime graph H.


Generalized matching Parametrized complexity 


  1. 1.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bodlaender, H.L., Fomin, F.V.: Equitable colorings of bounded treewidth graphs. Theoret. Comput. Sci. 349(1), 22–30 (2005). Graph Colorings 2003MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boesch, F.T., Gimpel, J.F.: Covering points of a digraph with point-disjoint paths and its application to code optimization. J. ACM 24(2), 192–198 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Diestel, R.: Graph Theory: Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2012)Google Scholar
  6. 6.
    Drucker, A.: New limits to classical and quantum instance compression. In: FOCS, vol. 2012, pp. 609–618 (2012)Google Scholar
  7. 7.
    Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Frank, A., Tardos, É.: An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gajarský, J., Lampis, M., Ordyniak, S.: Parameterized algorithms for modular-width. In: Gutin, G., Szeider, S. (eds.) IPEC 2013. LNCS, vol. 8246, pp. 163–176. Springer, Heidelberg (2013). doi: 10.1007/978-3-319-03898-8_15 CrossRefGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman & Co., New York (1979)zbMATHGoogle Scholar
  11. 11.
    Hansen, P., Hertz, A., Kuplinsky, J.: Bounded vertex colorings of graphs. Discrete Math. 111(1–3), 305–312 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hope, A.K.: Component placement through graph partitioning in computer-aided printed-wiring-board design. Electron. Lett. 8(4), 87–88 (1972)CrossRefGoogle Scholar
  13. 13.
    Jarvis, M., Zhou, B.: Bounded vertex coloring of trees. Discrete Math. 232(1–3), 145–151 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lenstra Jr., H.W.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kirkpatrick, D.G., Hell, P.: On the completeness of a generalized matching problem. In: STOC 1978, pp. 240–245. ACM, New York (1978)Google Scholar
  16. 16.
    Lampis, M.: Algorithmic meta-theorems for restrictions of treewidth. Algorithmica 64(1), 19–37 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Libkin, L.: Elements of Finite Model Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg (2004)CrossRefzbMATHGoogle Scholar
  18. 18.
    Rao, M.: MSOL partitioning problems on graphs of bounded treewidth and clique-width. Theor. Comput. Sci. 377(1–3), 260–267 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tedder, M., Corneil, D., Habib, M., Paul, C.: Simpler linear-time modular decomposition via recursive factorizing permutations. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5125, pp. 634–645. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-70575-8_52 CrossRefGoogle Scholar
  20. 20.
    Tutte, W.T.: The factorization of linear graphs. J. Lond. Math. Soc. s1–22, 107–111 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    van Bevern, R., Bredereck, R., Bulteau, L., Chen, J., Froese, V., Niedermeier, R., Woeginger, G.J.: Star partitions of perfect graphs. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 174–185. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-43948-7_15 Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsCharles UniversityPragueCzech Republic

Personalised recommendations