Substitution decomposition on chordal graphs and applications

  • Wen-Lian Hsu
  • Tze-Heng Ma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 557)


In this paper, we present a linear time algorithm for substitution decomposition on chordal graphs. Based on this result, we develop a linear time algorithm for transitive orientation on chordal comparability graphs. Which reduces the complexity of chordal comparability recognition from O(n2) to O(n+m). We also devise a simple linear time algorithm for interval graph recognition where no complicated data structure is involved.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    K. S. Booth and G. S. Lueker, “Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-tree Algorithms,” J. Comput. Syst. Sci., v. 13, 1976, pp. 335–379.Google Scholar
  2. [2]
    D. Coppersmith and S. Winograd, “Matrix Multiplication via Arithmetic Progressions,” Proceedings of the 19th Annual Symposium on the Theory of Computation, 1987, pp. 1–6.Google Scholar
  3. [3]
    D. Duffus, I. Rival, and P. Winkler, “Minimizing Setups for Cycle-free Ordered Sets,” Proc. of the American Math. Soc., v. 85, 1982, pp. 509–513.Google Scholar
  4. [4]
    P. C. Fishburn, Interval Orders and Interval Graphs, Wiley, New York, 1985.Google Scholar
  5. [5]
    F. Gavril, “The Intersection Graphs of Subtrees in Trees are Exactly the Chordal Graphs,” J. Combin. Theory B, v. 16, 1974, pp. 47–56.Google Scholar
  6. [6]
    M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.Google Scholar
  7. [7]
    N. Korte and R. H. Möhring, “An Incremental Linear-Time Algorithm for Recognizing Interval Graphs,” SIAM J. Computing, v. 18, 1989, pp. 68–81.Google Scholar
  8. [8]
    W. L. Hsu, The Recognition and Isomorphism Problems for Circular-arc Graphs, preprint, 1989.Google Scholar
  9. [9]
    C. G. Lekkerkerker and J. Boland, “Representation of a Finite Graph by a Set of Intervals on the Real Line,” Fund. Math., v. 51, 1962, pp. 45–64.Google Scholar
  10. [10]
    J. H. Muller and J. Spinrad, “Incremental Modular Decomposition,” Journal of the ACM, v. 36, 1989, pp. 1–19.Google Scholar
  11. [11]
    T. H. Ma and J. Spinrad, “Cycle-free Partial Orders and Chordal Comparability Graphs,” Order, to appear.Google Scholar
  12. [12]
    D. J. Rose, R. E. Tarjan, and G. S. Lueker, “Algorithmic Aspects of Vertex Elimination of Graphs,” SIAM J. Comput., v. 5, 1976, pp. 266–283.Google Scholar
  13. [13]
    J. Spinrad, “On Comparability and Permutation Graphs,” SIAM J. Comput., v. 14, 1985, pp. 658–670.Google Scholar
  14. [14]
    J. Spinrad, “P4 Trees and Substitution Decomposition,” Discrete Applied Math., to appear, 1989.Google Scholar
  15. [15]
    R. E. Tarjan, “Amortized Computational Complexity,” SIAM J. Alg. Disc. Meth., v. 6, 1985, pp. 306–318.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Wen-Lian Hsu
    • 1
  • Tze-Heng Ma
    • 1
  1. 1.Institute of Information ScienceAcademia SinicaTaipeiRepublic of China

Personalised recommendations