Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width

(Extended Abstract)
  • B. Courcelle
  • J. A. Makowsky
  • U. Rotics
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1517)


Graphs of clique-width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we show that the (q,q-4) graphs are of clique width at most q and P4-tidy graphs are of clique-width at most 4. Furthermore, the k-expression (for k=4 or k=q) associated with such a graph can be found in linear time.

q,q-4) graphs were introduced by Babel and Olariu (1995) and extends the class of P4-sparse graphs. P4-sparse graphs were introduced by Hoàng (1985) and are widely studied because of their applications in areas such as scheduling, clustering and computational semantics. Another extension of P4-sparse graphs are the P4-tidy graphs which were introduced by Rusu (1995).

Furthermore, we show that the class of LinEMSOL(τ1,L) optimization problems is solvable in O(f(|V|,|E|)) time on a class of graphs of clique-width at most k in which for every graph G an expression defining it can be constructed in O(f(|V|,|E|)) time. By the above this applies in particular to (q,q – 4) graphs, P4-tidy graphs and P4-sparse graphs with f linear.

Finally, we show that the above results cannot be extended to MSOL(τ2) decision and optimization problems on the vocabulary τ2 which allow edges to be considered as elements of the domains of the graphs in question, and by that, allow quantifying over edges in addition to quantifying over vertices.


Internal Node Steiner Tree Prime Graph Graph Grammar Unary Predicate 
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. Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree decomposable graphs. Journal of Algorithms 12, 308–340 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Buer, H., Möhring, R.H.: A fast algorithm for the decomposition of graphs and posets. Math. Oper. Res. 8, 170–184 (1983)Google Scholar
  3. Babel, L., Olariu, S.: On the isomorphism of graphs with few P 4s. In: Nagl, M. (ed.) WG 1995. LNCS, vol. 1017, pp. 24–36. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  4. Babel, L., Olariu, S.: Domination and steiner tree problems on graphs with few P4’s. In: Hromkovič, J., Sýkora, O. (eds.) WG 1998. LNCS, vol. 1517, pp. 337–350. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  5. Courcelle, B., Engelfriet, J., Rozenberg, G.: Handle-rewriting hypergraph grammars. J. Comput. System Sci. 46, 218–270 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Cournier, A., Habib, M.: A new linear algorithm for modular decomposition. LNCS, vol. 787, pp. 68–84 (1994)Google Scholar
  7. Courcelle, B., Mosbah, M.: Monadic second-order evaluations on tree-decomposable graphs. Theoretical Computer Science  109, 49–82 (1993)Google Scholar
  8. Courcelle, B., Olariu, S.: Upper bounds to the clique-width of graphs (submitted for publication),
  9. Courcelle, B.: The monadic second-order logic of graphs i: Recognizable sets of finite graphs. Information and Computation 85, 12–75 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Courcelle, B.: The monadic second-order logic of graphs V: On closing the gap between definability and recognizability. Theoret. Comput. Sci. 80, 153–202 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Courcelle, B.: The monadic second-order logic of graphs VI: On several representations of graphs by relational structures. Disc. Appl. Math. 54, 117–149 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Courcelle, B.: The monadic second-order logic of graphs VIII: Orientations. Annals Pure Applied Logic 72, 103–143 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Courcelle, B.: The monadic second-order logic of graphs X: Linear orders. Theoret. Comput. Sci. 160, 87–143 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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, Ch. 5, vol. 1, pp. 313–400. World Scientific, Singapore (1997)CrossRefGoogle Scholar
  15. Ebbinghaus, H.D., Flum, J.: Finite Model Theory. Perspectives in Mathematical Logic. Springer, Heidelberg (1995)zbMATHGoogle Scholar
  16. Ehrenfeucht, A.: An application of games to the completeness problem for formalized theories. Fundamenta Mathematicae 49, 129–141 (1961)MathSciNetzbMATHGoogle Scholar
  17. Fagin, R.: Generalized first-order spectra and polynomial time recognizable sets. American Math. Society Proc. 7, 27–41 (1974)MathSciNetzbMATHGoogle Scholar
  18. Feferman, S.: Some recent work of Ehrenfeucht and Fraïssé. In: Proceedings of the Summer Institute of Symbolic Logic, Ithaca, pp. 201–209 (1957)Google Scholar
  19. Feferman, S., Vaught, R.: The first order properties of algebraic systems. Fundamenta Mathematicae 47, 57–103 (1959)MathSciNetzbMATHGoogle Scholar
  20. Garey, M.G., Johnson, D.S.: Computers and Intractability. Mathematical Series. W.H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  21. Giakoumakis, V., Roussel, F., Thuillier, H.: On P4-tidy graphs. Discrete Mathematics and Theoretical Computer Science 1, 17–41 (1997)MathSciNetzbMATHGoogle Scholar
  22. Gurevich, Y.: Modest theory of short chains. I. Journal of Symbolic Logic 44, 481–490 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Gurevich, Y.: Monadic second order theories. In: Model-Theoretic Logics, Perspectives in Mathematical Logic, Ch. 14. Springer, Heidelberg (1985)Google Scholar
  24. Giakoumakis, V., Vanherpe, J.: On extended P 4-reducible and extended P4-sparse graphs. Theoret. Comput. Sci. 180, 269–286 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Hoàng, C.: Doctoral thesis. McGill University, Montreal (1985)Google Scholar
  26. Jamison, B., Olariu, S.: P4-reducible graphs a class of tree representable graphs. Studies Appl. Math. 81, 79–87 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Jamison, B., Olariu, S.: A linear-time recognition algorithm for P4-sparse graphs. SIAM J. Comput. 21, 381–406 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Jamison, B., Olariu, S.: A unique tree representation for P4-sparse graphs. Discrete Appl. Math. 35, 115–129 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Jamison, B., Olariu, S.: A linear-time algorithm to recognize P4-reducible graphs. Theoret. Comput. Sci. 145, 329–344 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Jamison, B., Olariu, S.: Linear-time optimization algorithms for P4-sparse graphs. Discrete Appl. Math. 61, 155–175 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Läuchli, H.: A decision procedure for the weak second order theory of linear order. In: Logic Colloquium 1966, pp. 189–197. North Holland, Amsterdam (1968)Google Scholar
  32. Lautemann, C.: CSL 1992. LNCS, vol. 702, pp. 327–339. Springer, Heidelberg (1993)zbMATHGoogle Scholar
  33. Papadimitriou, C.: Computational Complexity. Addison Wesley, Reading (1994)zbMATHGoogle Scholar
  34. Rotics, U.: Efficient Algorithms for Generally Intractable Graph Problems Restricted to Specific Classes of Graphs. PhD thesis, Technion- Israel Institute of Technology (1998)Google Scholar
  35. Shelah, S.: The monadic theory of order. Annals of Mathematics 102, 379–419 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  36. Spinrad, J.: P4 -trees and substitution decomposition. Discrete Appl. Math. 39, 263–291 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  37. Wanke, E.: k-NLC graphs and polynomial algorithms. Discrete Appl. Math. 54, 251–266 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • B. Courcelle
    • 1
  • J. A. Makowsky
    • 2
  • U. Rotics
    • 2
  1. 1.Laboratoire d’InformatiqueUniversité Bordeaux-ITalenceFrance
  2. 2.Department of Computer ScienceTechnion, Israel Institute of TechnologyHaifaIsrael

Personalised recommendations