Solving Problems on Graphs of High Rank-Width

  • Eduard Eiben
  • Robert Ganian
  • Stefan Szeider
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)


A modulator of a graph G to a specified graph class \({\mathcal H}\) is a set of vertices whose deletion puts G into \({\mathcal H}\). The cardinality of a modulator to various graph classes has long been used as a structural parameter which can be exploited to obtain FPT algorithms for a range of hard problems. Here we investigate what happens when a graph contains a modulator which is large but “well-structured” (in the sense of having bounded rank-width). Can such modulators still be exploited to obtain efficient algorithms? And is it even possible to find such modulators efficiently?

We first show that the parameters derived from such well-structured modulators are strictly more general than the cardinality of modulators and rank-width itself. Then, we develop an FPT algorithm for finding such well-structured modulators to any graph class which can be characterized by a finite set of forbidden induced subgraphs. We proceed by showing how well-structured modulators can be used to obtain efficient parameterized algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use the concept of well-structured modulators to develop an algorithmic meta-theorem for efficiently deciding problems expressible in Monadic Second Order (MSO) logic, and prove that this result is tight in the sense that it cannot be generalized to LinEMSO problems.


Vertex Cover Maximum Clique Graph Class Split Graph Minimum Vertex Cover 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Alekseev, V.E.: Polynomial algorithm for finding the largest independent sets in graphs without forks. Discr. Appl. Math. 135(1–3), 3–16 (2004)Google Scholar
  2. 2.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Kernel bounds for path and cycle problems. Theor. Comput. Sci. 511, 117–136 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brandstädt, A., Lozin, V.V.: A note on alpha-redundant vertices in graphs. Discr. Appl. Math. 108(3), 301–308 (2001)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cai, L.: Parameterized complexity of vertex colouring. Discr. Appl. Math. 127(3), 415–429 (2003)CrossRefzbMATHGoogle Scholar
  5. 5.
    Corneil, D.G., Lerchs, H., Burlingham, L.S.: Complement reducible graphs. Discr. Appl. Math. 3, 163–174 (1981)MathSciNetzbMATHGoogle Scholar
  6. 6.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cunningham, W.H.: Decomposition of directed graphs. SIAM J. Algebraic Discrete Methods 3(2), 214–228 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Diestel, R.: Graph Theory. GTM, vol. 173, 2nd edn. Springer Verlag, New York (2000)Google Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer Verlag (2013)Google Scholar
  10. 10.
    Gajarský, J., Hliněný, P., Obdržálek, J., Ordyniak, S., Reidl, F., Rossmanith, P., Villaamil, F.S., Sikdar, S.: Kernelization using structural parameters on sparse graph classes. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 529–540. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  11. 11.
    Ganian, R., Hliněný, P.: On parse trees and Myhill-Nerode-type tools for handling graphs of bounded rank-width. Discr. Appl. Math. 158(7), 851–867 (2010)CrossRefzbMATHGoogle Scholar
  12. 12.
    Gaspers, S., Misra, N., Ordyniak, S., Szeider, S., Živný, S.: Backdoors into heterogeneous classes of SAT and CSP. In: Brodley, C.E., Stone, P.(eds.), Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, pp. 2652–2658. AAAI Press (2014)Google Scholar
  13. 13.
    Gerber, M.U., Lozin, V.V.: Robust algorithms for the stable set problem. Graphs and Combinatorics 19(3), 347–356 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gioan, E., Paul, C.: Dynamic distance hereditary graphs using split decomposition. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 41–51. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  15. 15.
    Gioan, E., Paul, C.: Split decomposition and graph-labelled trees: characterizations and fully dynamic algorithms for totally decomposable graphs. Discr. Appl. Math. 160(6), 708–733 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gioan, E., Paul, C., Tedder, M., Corneil, D.: Practical and efficient split decomposition via graph-labelled trees. Algorithmica 69(4), 789–843 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Golovach, P.A., Paulusma, D., Song, J.: Closing complexity gaps for coloring problems on h-free graphs. Inf. Comput. 237, 204–214 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hliněný, P., Oum, S.I.: Finding branch-decompositions and rank-decompositions. SIAM J. Comput. 38(3), 1012–1032 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kochol, M., Lozin, V.V., Randerath, B.: The 3-colorability problem on graphs with maximum degree four. SIAM J. Comput. 32(5), 1128–1139 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Libkin, L.: Elements of Finite Model Theory. Springer (2004)Google Scholar
  21. 21.
    Lokshantov, D., Vatshelle, M., Villanger, Y.: Independent set in p\(_{\text{5 }}\)-free graphs in polynomial time. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, pp. 570–581. SIAM (2014)Google Scholar
  22. 22.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2006) CrossRefzbMATHGoogle Scholar
  23. 23.
    Oum, S., Seymour, P.: Approximating clique-width and branch-width. J. Combin. Theory Ser. B 96(4), 514–528 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Algorithms and Complexity Group, Institute of Computer Graphics and AlgorithmsTU WienViennaAustria

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