Graph Structure and Monadic Second-Order Logic: Language Theoretical Aspects

  • Bruno Courcelle
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5125)


Graph structure is a flexible concept covering many different types of graph properties. Hierarchical decompositions yielding the notions of tree-width and clique-width, expressed by terms written with appropriate graph operations and associated with Monadic Second-order Logic are important tools for the construction of Fixed-Parameter Tractable algorithms and also for the extension of methods and results of Formal Language Theory to the description of sets of finite graphs. This informal overview presents the main definitions, results and open problems and tries to answer some frequently asked questions.


Graph Structure Graph Transformation Derivation Tree Graph Grammar Circle Graph 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Blumensath, A., Colcombet, T., Löding, C.: Logical Theories and Compatible Operations. In: Flum, J., Grädel, E., Wilke, T. (eds.) Logic and automata: History and Perspectives, pp. 73–106. University Press, Amsterdam (2008)Google Scholar
  2. 2.
    Courcelle, B.: The Expression of Graph Properties and Graph Transformations in Monadic Second-Order Logic. In: Rozenberg, G. (ed.) Handbook of Graph Grammars and Computing by Graph Transformations. Foundations, vol. 1, pp. 313–400. World Scientific, Singapore (1997)CrossRefGoogle Scholar
  3. 3.
    Courcelle, B.: Graph Structure and Monadic Second-order Logic. Cambridge University Press, Cambridge (in preparation),
  4. 4.
    Downey, R., Fellows, M.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  5. 5.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (to appear)Google Scholar
  6. 6.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  7. 7.
    Grohe, M.: Logic, Graphs, and Algorithms. In: Flum, J., Grädel, E., Wilke, T. (eds.) Logic and automata: History and Perspectives, pp. 357–422. Amsterdam University Press (2008)Google Scholar
  8. 8.
    Hell, P., Nešetřil, J.: Graphs and homomorphisms. Oxford University Press, Oxford (2004)CrossRefzbMATHGoogle Scholar
  9. 9.
    Makowsky, J.: Algorithmic uses of the Feferman-Vaught Theorem. Ann. Pure Appl. Logic 126, 159–213 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rozenberg, G.: Handbook of Graph Grammars and Computing by Graph Transformations. Foundations, vol. 1. World Scientific, Singapore (1997)CrossRefzbMATHGoogle Scholar
  11. 11.
    Tutte, W.: Graph Theory. Addison–Wesley, Reading (1984)zbMATHGoogle Scholar
  12. 12.
    Blumensath, A., Courcelle, B.: Recognizability, Hypergraph Operations, and Logical Types. Inf.Comput. 204, 853–919 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bodlaender, H.: A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth. SIAM J. Comput. 25, 1305–1317 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bodlaender, H.: Treewidth: Characterizations, Applications, and Computations. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 1–14. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Brandstädt, A., Dragan, F., Le, H., Mosca, R.: New Graph Classes of Bounded Clique-Width. Theory Comput. Syst. 38, 623–645 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Brandstädt, A., Engelfriet, J., Le, H., Lozin, V.: Clique-Width for 4-Vertex Forbidden Subgraphs. Theory Comput. Syst. 39, 561–590 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: Progress on Perfect Graphs. Mathematical programming, Ser. B 97, 405–422 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Courcelle, B., Oum, S.: Vertex-minors, Monadic Second-Order Logic, and a Conjecture by Seese. J. Comb. Theory, Ser. B 97, 91–126 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Courcelle, B.: The Monadic Second-Order Logic of Graphs VII: Graphs as Relational Structures. Theor. Comput. Sci. 101, 3–33 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Courcelle, B.: The Monadic Second-Order Logic of Graphs X: Linear Orderings. Theor. Comput. Sci. 160, 87–143 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Courcelle, B.: The Monadic Second-Order Logic of Graphs XI: Hierarchical Decompositions of Connected Graphs. Theor. Comput. Sci. 224, 35–58 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Courcelle, B.: The Monadic Second-Order Logic of Graphs XII: Planar Graphs and Planar Maps. Theor. Comput. Sci. 237, 1–32 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Courcelle, B.: The Monadic Second-Order Logic of Graphs XIV: Uniformly Sparse Graphs and Edge Set Quantifications. Theor. Comput. Sci. 299, 1–36 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Courcelle, B.: The Monadic Second-Order Logic of Graphs XV: On a conjecture by D. Seese. J. Applied Logic 4, 79–114 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Courcelle, B.: The Monadic Second-Order Logic of Graphs XVI: Canonical graph decompositions. Logical Methods in Computer Science 2 (2006)Google Scholar
  26. 26.
    Courcelle, B.: Circle Graphs and Monadic Second-order logic. Journal of Applied Logic (in press)Google Scholar
  27. 27.
    Courcelle, B., Engelfriet, J.: A Logical Characterization of the Sets of Hypergraphs Defined by Hyperedge Replacement Grammars. Mathematical Systems Theory 28, 515–552 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Courcelle, B., Engelfriet, J., Rozenberg, G.: Handle-Rewriting Hypergraph Grammars. J. Comput. Syst. Sci. 46, 218–270 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Courcelle, B., Makowsky, J.: Fusion in Relational Structures and the Verification of Monadic Second-Order Properties. Mathematical Structures in Computer Science 12, 203–235 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Courcelle, B., Makowsky, J., Rotics, U.: Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width. Theory Comput. Syst. 33, 125–150 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Courcelle, B., Weil, P.: The Recognizability of Sets of Graphs is a Robust Property. Theor. Comput. Sci. 342, 173–228 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Courcelle, B., Walukiewicz, I.: Monadic Second-Order Logic, Graph Coverings and Unfoldings of Transition Systems. Ann. Pure Appl. Logic 92, 35–62 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Cunnigham, W.: Decomposition of Directed Graphs. SIAM Algor. Discrete Meth. 3, 214–228 (1982)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Engelfriet, J., van Oostrom, V.: Logical Description of Contex-Free Graph Languages. J. Comput. Syst. Sci. 55, 489–503 (1997)CrossRefzbMATHGoogle Scholar
  35. 35.
    Fellows, M., Rosamond, F., Rotics, U., Szeider, S.: Clique-width Minimization is NP-hard. In: 38th Annual ACM Symposium on Theory of Computing, pp. 354–362 (2006)Google Scholar
  36. 36.
    Frick, M.: Generalized Model-Checking over Locally Tree-Decomposable Classes. Theor. Comput. Sci. 37, 157–191 (2004)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Frick, M., Grohe, M.: The Complexity of First-order and Monadic second-order Logic Revisited. Ann. Pure Appl. Logic 130, 3–31 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Gallai, T.: Transitiv Orientierbare Graphen. Acta Math. Acad. Sci. Hungar 18, 25–66 (1967); Translation in English by Maffray, F. Preissmann, M.: In: Ramirez Alfonsin,J.L., Reed, B.A.: (eds.), Perfect Graphs, pp. 25-66, Wiley, New York (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Grädel, E., Hirsch, C., Otto, M.: Back and Forth Between Guarded and Modal Logics. ACM Trans. Comput. Log. 3, 418–463 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Lapoire, D.: Recognizability Equals Monadic Second-Order Definability for Sets of Graphs of Bounded Tree-Width. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 618–628. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  41. 41.
    Hlinený, P., Oum, S.: Finding Branch-Decompositions and Rank-Decompositions. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 163–174. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  42. 42.
    Klarlund, N.: Mona & Fido: The Logic-Automaton Connection in Practice. In: Nielsen, M. (ed.) CSL 1997. LNCS, vol. 1414, pp. 311–326. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  43. 43.
    Madelaine, F.: Universal Structures and the Logic of Forbidden Patterns. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, pp. 471–485. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  44. 44.
    Makowsky, J., Marino, J.: Tree-width and the Monadic Quantifier Hierarchy. Theor. Comput. Sci. 303, 157–170 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Makowsky, J., Pnueli, Y.: Arity and Alternation in Second-Order Logic. Ann. Pure Appl. Logic 78, 189–202 (1996); Erratum: Ann. Pure Appl. Logic 92, 215 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Mezei, J., Wright, J.: Algebraic Automata and Context-Free Sets. Information and Control 11, 3–29 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Nešetřil, J., de Mendez, P.O.: Linear Time Low Tree-width Partitions and Algorithmic Consequences. In: Proc. Symp. Theory of Computation, pp. 391–400 (2006)Google Scholar
  48. 48.
    Oum, S.: Rank-width and Vertex-minors. J. Comb. Theory, Ser. B 95, 79–100 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Oum, S., Seymour, P.: Approximating Clique-width and Branch-width. J. Comb. Theory, Ser. B 96, 514–528 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Robertson, N., Seymour, P.: Graph Minors. V. Excluding a Planar Graph. J. Comb. Theory, Ser. B 41, 92–114 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Robertson, N., Seymour, P.: Graph minors. VIII. A Kuratowski Theorem for General Surfaces. J. Comb. Theory, Ser. B 48, 255–288 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Robertson, N., Seymour, P.: Graph Minors. XVI. Excluding a Non-planar Graph. J. Comb. Theory, Ser. B 89, 43–76 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Seese, D.: The Structure of Models of Decidable Monadic Theories of Graphs. Ann. Pure Appl. Logic 53, 169–195 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Soguet, D.: Génération Automatique d’Algorithmes Linéaires, Doctoral dissertation, Paris-Sud University, France (July 2008)Google Scholar
  55. 55.
    Wanke, E.: k-NLC Graphs and Polynomial Algorithms. Discrete Applied Mathematics 54, 251–266 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Bruno Courcelle
    • 1
  1. 1.Institut Universitaire de FranceUniversité Bordeaux-1, LaBRI, CNRSTalence cedexFrance

Personalised recommendations