Monadic NP and Graph Minors

  • Martin Kreidler
  • Detlef Seese
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1584)


In our paper, we prove that Graph Connectivity is not in Monadic NP even in the presence of a built-in relation of arbitrary degree that does not have for an arbitrary, but fixed k ≥ 2 ∈ IN the complete graph K k as a minor. We obtain our result by using the method of indiscernibles and giving a winning strategy for the duplicator in the Ajtai-Fagin Ehrenfeucht-Fraïssé Game .

The result is afterwards strengthened to arbitrary forbidden minors and to minor-closed classes of binary relations.


Monadic Second-Order Logic Descriptive Complexity Theory Ehrenfeucht-Fraïssé Games Finite Model Theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ajtai, M., Fagin, R.: Reachability is harder for directed than for undirected finite graphs. Journal of Symbolic Logic 55(1), 113–150 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arnborg, S., Lagergren, J., Seese, D.: Easy Problems for Tree-Decomposable Graphs. Journal of Algorithms 12, 308–340 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chang, C., Keisler, H.: Model Theory, 3rd edn. North-Holland, Amsterdam (1974) (1990)Google Scholar
  4. 4.
    Diestel, R.: Graph Theory. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  5. 5.
    Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. Springer, Heidelberg (1995)zbMATHGoogle Scholar
  6. 6.
    Fagin, R.: Generalized First-Order spectra and polynomial-time recognizable sets. In: Karp, R. (ed.) Complexity of Computation, SIAM-AMS Proc., vol. 7, pp. 27–41 (1974)Google Scholar
  7. 7.
    Fagin, R.: Monadic generalized spectra. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 21, 89–96 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Fagin, R.: Comparing the Power of Games on Graphs. Mathematical Logic Quarterly 43, 431–455 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fagin, R., Stockmeyer, L., Vardi, M.: On Monadic NP vs. Monadic co-NP. Information and Computation 120(1), 78–92 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993)zbMATHCrossRefGoogle Scholar
  11. 11.
    Immerman, N.: Languages That Capture Complexity Classes. SIAM Journal of Computing 16(4), 760–778 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kreidler, M., Seese, D.: Monadic NP and Built-in Trees. In: van Dalen, D., Bezem, M. (eds.) CSL 1996. LNCS, vol. 1258, pp. 260–274. Springer, Heidelberg (1997)Google Scholar
  13. 13.
    Papadimitriou, C.H.: Computational Complexity. Addison-Wesley Publishing Company, Reading (1994)zbMATHGoogle Scholar
  14. 14.
    Robertson, N., Seymour, P.: Graph minors - a survey. In: Anderson, I. (ed.) Surveys in Combinatorics, pp. 153–171. Cambridge University Press, Cambridge (1985)Google Scholar
  15. 15.
    Robertson, N., Seymour, P.D.: Graph Minors IV: Tree-Width and Well-Quasi- Ordering. Journal of Comb. Theory Series B 48, 227–254 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Robertson, N., Seymour, P.: Graph minors XII: Excluding a non-planar graph. Journal of Combinatorial Theory Series B 64, 240–272 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Robertson, N., Seymour, P.: Graph minors XIII: The disjoint path problem. Journal of Combinatorial Theory Series B 63, 65–110 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Schwentick, T.: On Winning Strategies in Ehrenfeucht Games and Monadic NP. Annals of Pure and Applied Logic 79, 61–92 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Seese, D., Wessel, W.: Graphminoren und Gitter: Zu einigen Arbeiten von N. Robertson und P. Seymour. In: Wagner, K., Bodendiek,R (eds.) Graphentheorie III, BI Wissenschaftsverlag (1993)Google Scholar
  20. 20.
    Shelah, S.: Classification Theory and the Number of Non-Isomorphic Models. Studies in Logic series. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  21. 21.
    Wagner, K.: Über eine Eigenschaft der ebenen Komplexe. Mathematische Annalen 114, 570–590 (1937)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Martin Kreidler
    • 1
  • Detlef Seese
    • 1
  1. 1.Institut AIFBUniversität KarlsruheKarlsruheGermany

Personalised recommendations