NC-algorithms for graphs with small treewidth

  • Hans L. Bodlaender
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 344)


In this paper we give a parallel algorithm for recognizing graphs with treewidth ≤ k, for constant k, and building the corresponding tree-decomposition, that uses O(log n) time and O(n3k+4) processors on a CRCW PRAM. Also, we give a parallel algorithm that transforms a given tree-decomposition of a graph G with treewidth k to another tree-decomposition of G with treewidth ≤ 3k+2, such that the tree in this tree-decomposition is binary and has logarithmic depth. The algorithm uses a linear number of processors and O(log n) time. Many NP-complete graph problems are known to be solvable in polynomial time, when restricted to graphs with treewidth ≤ k, k constant. From the results in this paper, it follows that most of these problems are also in NC, when restricted to graphs with treewidth bounded by a constant.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Arnborg. Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey. BIT, 25:2–23, 1985.Google Scholar
  2. [2]
    S. Arnborg, D. G. Corneil, and A. Proskurowski. Complexity of finding embeddings in a k-tree. SIAM J. Alg. Disc. Meth., 8:277–284, 1987.Google Scholar
  3. [3]
    S. Arnborg, J. Lagergren, and D. Seese. Problems easy for tree-decomposable graphs (extended abstract). In Proc. 15 th ICALP, pages 38–51, Springer Verlag, Lect. Notes in Comp. Sc. 317, 1988.Google Scholar
  4. [4]
    S. Arnborg and A. Proskurowski. Linear time algorithms for NP-hard problems on graphs embedded in k-trees. TRITA-NA-8404, Dept. of Num. Anal. and Comp. Sci., Royal Institute of Technology, Stockholm, Sweden, 1984.Google Scholar
  5. [5]
    H. L. Bodlaender. Classes of Graphs with Bounded Treewidth. Technical Report RUU-CS-86-22, Dept. Of Comp. Science, University of Utrecht, Utrecht, 1986.Google Scholar
  6. [6]
    H. L. Bodlaender. Dynamic programming algorithms on graphs with bounded tree-width. Tech. Rep., Lab. for Comp. Science, M.I.T., 1987. Extended abstract in proceedings ICALP 88.Google Scholar
  7. [7]
    H. L. Bodlaender. Polynomial algorithms for Graph Isomorphism and Chromatic Index on partial k-trees. In Proc. 1st Scandinavian Workshop on Algorithm Theory, pages 223–232, Springer Verlag LNCS 318, 1988.Google Scholar
  8. [8]
    N. Chandrasekharan and S. S. Iyengar. NC Algorithms for Recognizing Chordal Graphs and k-Trees. Tech. Rep. 86-020, Dept. of Comp. Science, Louisiana State University, 1986.Google Scholar
  9. [9]
    B. Courcelle. Recognizability and Second-Order Definability for Sets of Finite Graphs. Preprint, Universite de Bordeaux, 1987.Google Scholar
  10. [10]
    J. Engelfriet, G. Leih, and E. Welzl. Characterization and complexity of boundary graph languages. 1987. Manuscript.Google Scholar
  11. [11]
    M. R. Fellows and M. A. Langston. On seach, decision and the efficiency of polynomial-time algorithms. 1988. Extended abstract.Google Scholar
  12. [12]
    M. R. Garey and D. S. Johnson. Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York, 1979.Google Scholar
  13. [13]
    A. M. Gibbons, A. Israeli, and W. Rytter. Parallel o(log n) time edge-coloring of trees and halin graphs. Inform. Proc. Letters, 27:43–52, 1988.Google Scholar
  14. [14]
    G. Miller and J. Reif. Parallel tree contraction and its application. In Proc. of the 26th Annual IEEE Symp. on the Foundations of Comp. Science, pages 478–489, 1985.Google Scholar
  15. [15]
    N. Robertson and P. Seymour. Graph minors. II. Algorithmic aspects of tree-width. J. of Algorithms, 7:309–322, 1986.Google Scholar
  16. [16]
    N. Robertson and P. Seymour. Graph minors. XIII. The disjoint paths problem. 1986. Manuscript.Google Scholar
  17. [17]
    P. Scheffler. Linear-time algorithms for NP-complete problems restricted to partial k-trees. Report R-MATH-03/87, Karl-Weierstrass-Institut Für Mathematik, Berlin, GDR, 1987.Google Scholar
  18. [18]
    P. Scheffler and D. Seese. A combinatorial and logical approach to linear-time computability. 1986. Extended abstract.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Hans L. Bodlaender
    • 1
  1. 1.Department of Computer ScienceUniversity of UtrechtUtrechtthe Netherlands

Personalised recommendations