Linear-Time Algorithms for Graphs of Bounded Rankwidth: A Fresh Look Using Game Theory

(Extended Abstract)
  • Alexander Langer
  • Peter Rossmanith
  • Somnath Sikdar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6648)


We present an alternative proof of a theorem by Courcelle, Makowski and Rotics [6] which states that problems expressible in MSO1 are solvable in linear time for graphs of bounded rankwidth. Our proof uses a game-theoretic approach and has the advantage of being self-contained. In particular, our presentation does not assume any background in logic or automata theory. Moreover our approach can be generalized to prove other results of a similar flavor, for example, that of Courcelle’s Theorem for treewidth [3,19].


Characteristic Tree Free Variable Parse Tree Winning Strategy Relation Symbol 
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  1. 1.
    Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12(2), 308–340 (1991)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Berlekamp, E.R., Conway, J.H., Guy, R.K.: Winning Ways for Your Mathematical Plays. A.K. Peters, Wellesley (1982)MATHGoogle Scholar
  3. 3.
    Courcelle, B.: The monadic second order theory of Graphs I: Recognisable sets of finite graphs. Information and Computation 85, 12–75 (1990)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Courcelle, B.: Monadic second-order definable graph transductions: A survey. Theor. Comput. Sci. 126(1), 53–75 (1994)CrossRefMATHGoogle Scholar
  5. 5.
    Courcelle, B., Kanté, M.M.: Graph operations characterizing rank-width and balanced graph expressions. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 66–75. Springer, Heidelberg (2007)CrossRefGoogle 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, 125–150 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Courcelle, B., Makowsky, J.A., Rotics, U.: On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discrete Applied Mathematics 108(1-2), 23–52 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Courcelle, B., Mosbah, M.: Monadic second-order evaluations on tree-decomposable graphs. Theor. Comput. Sci. 109(1-2), 49–82 (1993)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. Springer, Heidelberg (1999)MATHGoogle Scholar
  10. 10.
    Feferman, S., Vaught, R.: The first order properties of algebraic systems. Fund. Math. 47, 57–103 (1959)MathSciNetMATHGoogle Scholar
  11. 11.
    Ganian, R., Hliněený, P.: On parse trees and Myhill–Nerode–type tools for handling graphs of bounded rank-width. Disc. App. Math. 158(7), 851–867 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ganian, R., Hliněný, P., Obdržálek, J.: Unified approach to polynomial algorithms on graphs of bounded (bi-)rank-width (2009) (submitted)Google Scholar
  13. 13.
    Grädel, E.: Finite model theory and descriptive complexity. In: Finite Model Theory and Its Applications, pp. 125–230. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Gurevich, Y.: Modest Theory of Short Chains. I. J. Symb. Log. 44(4), 481–490 (1979)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gurevich, Y.: Monadic second-order theories. In: Jon Barwise, S.F. (ed.) Model-Theoretic Logics, pp. 479–506. Springer, Heidelberg (1985)Google Scholar
  16. 16.
    Hintikka, J.: Logic, Language-Games and Information: Kantian Themes in the Philosophy of Logic. Clarendon Press, Oxford (1973)MATHGoogle Scholar
  17. 17.
    Hliněný, P., Oum, S.: Finding branch-decomposition and rank-decomposition. SIAM Journal on Computing 38, 1012–1032 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kante, M.M.: The rankwidth of directed graphs (2007) (preprint),
  19. 19.
    Kneis, J., Langer, A., Rossmanith, P.: Courcelle’s Theorem – a game-theoretic approach (2010) (submitted)Google Scholar
  20. 20.
    Oum, S.: Graphs of Bounded Rankwidth. PhD thesis, Princeton University (2005)Google Scholar
  21. 21.
    Oum, S., Seymour, P.D.: Approximating clique-width and branch-width. Journal of Combinatorial Theory Series B 96(4), 514–528 (2006)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Øverlier, L., Syverson, P.: Locating hidden servers. In: Proceedings of the 2006 IEEE Symposium on Security and Privacy. IEEE CS, Los Alamitos (May 2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Alexander Langer
    • 1
  • Peter Rossmanith
    • 1
  • Somnath Sikdar
    • 1
  1. 1.RWTH Aachen UniversityAachenGermany

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