An algebraic theory of graph reduction

  • Stefan Arnborg
  • Bruno Courcelle
  • Andrzej Proskurowski
  • Detlef Seese
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 532)


We show how membership in classes of graphs definable in monadic second order logic and of bounded treewidth can be decided by finite sets of terminating reduction rules. The method is constructive in the sense that we describe an algorithm which will produce, from a formula in monadic second order logic and an integer k such that the class defined by the formula is of treewidth ≤ k, a set of rewrite rules that reduces any member of the class to one of finitely many graphs, in a number of steps bounded by the size of the graph. This reduction system corresponds to an algorithm that runs in time linear in the size of the graph.


Graph algebra reduction 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A.V. Aho, J.E. Hopcroft and J.D. Ullman, Design and Analysis of Computer Algorithms Addison-Wesley 1972.Google Scholar
  2. [2]
    S. Arnborg, Efficient Algorithms for Combinatorial Problems on Graphs with Bounded Decomposability — A Survey, BIT 25 (1985), 2–33CrossRefGoogle Scholar
  3. [3]
    S. Arnborg, D.G. Corneil and A. Proskurowski, Complexity of Finding Embeddings in a k-tree, SIAM J. Alg. and Discr. Methods 8(1987), 277–287Google Scholar
  4. [4]
    S. Arnborg, B. Courcelle, A. Proskurowski, and D. Seese,An algebraic theory of graph reduction, Technical Report LaBRI TR 90-02, University of Bordeaux (1990).Google Scholar
  5. [5]
    S. Arnborg, J. Lagergren and D. Seese, Problems Easy for Tree-decomposable graphs (extended abstract). Proc. 15 th ICALP, Springer Verlag, Lect. Notes in Comp. Sc.317 (1988) 38–51Google Scholar
  6. [6]
    S. Arnborg, J. Lagergren and D. Seese, Problems Easy for Tree-decomposable graphs to appear, J. of Algorithms.Google Scholar
  7. [7]
    S. Arnborg and A. Proskurowski, Characterization and Recognition of Partial 3-trees, SIAM J. Alg. and Discr. Methods 7(1986), 305–314Google Scholar
  8. [8]
    S. Arnborg and A. Proskurowski, Linear Time Algorithms for NP-hard Problems on Graphs Embedded in k-trees, Discr. Appl. Math. 23(1989) 11–24CrossRefGoogle Scholar
  9. [9]
    S. Arnborg, A. Proskurowski and D.G. Corneil, Forbidden minors characterization of partial 3-trees, Discrete Math., to appearGoogle Scholar
  10. [10]
    M. Bauderon and B. Courcelle, Graph expressions and graph rewritings, Mathematical Systems Theory 20(1987), 83–127CrossRefGoogle Scholar
  11. [11]
    J.A. Bern, E. Lawler and A. Wong, Linear time computation of optimalsubgraphs of decomposable graphs, J. of Algorithms 8 (1987), 216–235CrossRefGoogle Scholar
  12. [12]
    T. Beyer, W. Jones and S. Mitchell, Linear algorithms for isomorphism of maximal outerplanar graphs, JACM 26(4), Oct.1979, 603–610CrossRefGoogle Scholar
  13. [13]
    H.L. Bodlaender, Dynamic Programming on Graphs with Bounded Tree-width, MIT/LCS/TR-394, MIT 1987.Google Scholar
  14. [14]
    H.L. Bodlaender, Improved self-reduction algorithms for graphs with bounded treewidth. RUU-CS-88-29, University of Utrecht 1988.Google Scholar
  15. [15]
    B. Courcelle, Equivalence and transformation of regular systems. Applications to recursive program schemes and grammars, Theoretical Computer Science 42(1986), 1–22CrossRefGoogle Scholar
  16. [16]
    B. Courcelle, The monadic second order logic of graphs I: Recognizable sets of finite graphs, Information and Computation85 (1990) 12–75CrossRefGoogle Scholar
  17. [17]
    B. Courcelle, The monadic second order logic of graphs III: Tree-width, forbidden minors, and complexity issues, Report I-8852, Bordeaux-1 University (1988)Google Scholar
  18. [18]
    B. Courcelle, Graph rewriting: an algebraic and logical approach, in “Handbook of Theoretical Computer Science”, Volume B, J.B. van Leeuwen, Ed. Elsevier, 194–242.Google Scholar
  19. [19]
    B. Courcelle, Some applications of logic, universal algebra and of category theory to the theory of graph transformations, Bulletin of the EATCS 36, October 1988, 161–218.Google Scholar
  20. [20]
    B. Courcelle, The monadic second-order logic of graphs: Definable sets of finite graphs, LNCS 344 (1989) 30–53.Google Scholar
  21. [21]
    B. Courcelle, Graphs as relational structures; an algebraic and logical approach, this volume.Google Scholar
  22. [22]
    J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, North Holland (1976)Google Scholar
  23. [23]
    H. Ehrig, M. Nagl, G. Rozenberg and A. Rosenfeld, (Eds.), Proceedings of the 3rd international workshop on Graph Grammars and their Application to Computer Science, Springer Verlag, Lect. Notes in Comp. Sc.291Google Scholar
  24. [24]
    M. Fellows and M. Langston, An analogue of the Myhill-Nerode theorem and its use in computing finite basis characterizations, FOCS 1989 520–525Google Scholar
  25. [25]
    M. Hecht and J. Ullmann, Flow graph reducibility, SIAM J. Comp. 1(1972)188–202CrossRefGoogle Scholar
  26. [26]
    J. Lagergren, manuscript (1987)Google Scholar
  27. [27]
    Y. Kajitani, A. Ishizuka and S. Ueno, Characterization of partial 3-trees in terms of 3 structures, Graphs and Combinatorics2(1986) 233–246.CrossRefGoogle Scholar
  28. [28]
    T. Lengauer and E. Wanke, Efficient analysis of graph properties on context-free graph languages, ICALP 88, LNCS 317, (1988), 379–393Google Scholar
  29. [29]
    J. Matoušek and R. Thomas, Algorithms finding tree-decompositions of graphs, manuscript (1988)Google Scholar
  30. [30]
    N. Robertson and P.D. Seymour, Some new results on the well-quasi ordering of graphs, Annals of Discrete Mathematics 23(1987), 343–354Google Scholar
  31. [31]
    N. Robertson and P.D. Seymour, Graph Minors X Preprint.Google Scholar
  32. [32]
    N. Robertson and P.D. Seymour, Graph Minors XIII, The Disjoint Path Problem Preprint.Google Scholar
  33. [33]
    N. Robertson and P.D. Seymour, Graph Minors XIV, Wagners conjecture, Preprint.Google Scholar
  34. [34]
    M.M. Syslo, Linear time Algorithm for Coding Outerplanar Graphs, Institute of Computer Science, Wroclaw University, Raport Nr N-20 (1977)Google Scholar
  35. [35]
    J.W. Thatcher, J.B. Wright, Generalized Finite Automata Theory with an Application to a Decision Problem in Second-Order Logic, Mathematical Systems Theory 2(1968), 57–81.CrossRefGoogle Scholar
  36. [36]
    A. Wald and C.J. Colbourn, Steiner Trees, Partial 2-trees, and Minimum IFI Networks, Networks 13 (1983), 159–167Google Scholar
  37. [37]
    T.V. Wimer, Linear algorithms on k-terminal graphs, PhD. thesis, Clemson University, August 1987Google Scholar
  38. [38]
    T.V.Wimer, S.T.Hedetniemi, and R.Laskar, A methodology for constructing linear graph algorithms, DCS, Clemson University, September 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Stefan Arnborg
    • 1
  • Bruno Courcelle
    • 2
  • Andrzej Proskurowski
    • 3
  • Detlef Seese
    • 4
  1. 1.The Royal Institute of Technology NADA, KTHStockholmSweden
  2. 2.Laboratorie d'Informatique (associé au CNRS)Bordeaux-1 UniversityTalenceFrance
  3. 3.CIS departmentUniversity of OregonEugeneUSA
  4. 4.Akademie der WissenschaftenKarl-Weierstraß Institute für MathematikBerlinGermany

Personalised recommendations