Measuring the distance to series-parallelity by path expressions

  • Valeska Naumann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 903)


Many graph and network problems are easily solved in the special case of series-parallel networks, but are highly intractable in the general case. This paper considers two complexity measures of two-terminal directed acyclic graphs (st-dags) describing the “distance” of an st-dag from series-parallelity. The two complexity measures are the factoring complexity ψ(G) and the reduction complexity μ(G). Bein, Kamburowski, and Stallmann [3] have shown that ψ(G)≤μ(G)≤n−3, where G is an st-dag with n nodes. They conjectured that ψ(G)=μ(G). This paper gives a proof for this conjecture.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Agrawal and A. Satyanarayana, An O(∥E∥) time algorithm for computing the reliability of a class of directed networks, Oper. Res., 32 (1984), pp. 493–515.Google Scholar
  2. 2.
    W. Bein, P. Brucker, and A. Tamir, Minimum cost flow algorithms for series parallel networks, Discrete Appl. Math., 10 (1985), pp. 117–124.Google Scholar
  3. 3.
    W. Bein, J. Kamburowski, and F. M. Stallmann, Optimal reductions of two-terminal directed acyclic graphs, SIAM J. Comput., 6 (1992), pp. 1112–1129.Google Scholar
  4. 4.
    G. Chartrand, D. Geller, and S. Hedetniemi, Graphs with forbidden subgraphs, J. Combinatorial Theory, 10 (1971), pp. 12–41.Google Scholar
  5. 5.
    R. Duffin, Topology of series-parallel networks, J. Math. Anal. Appl., 10 (1965), pp. 303–318.Google Scholar
  6. 6.
    D. G. Kirkpatrick, and P. Hell, On the completeness of a generalized matching problem, Proc. 10th ACM Symp. on Theory of Computing, San Diego, Calif., 1978, pp. 265–274Google Scholar
  7. 7.
    R. H. Möhring, Computationally tractable classes of ordered sets, In: I. Rival, ed., Algorithms and Order, Kluwer Acad. Publ.; Dordrecht, 1989, pp. 105–194Google Scholar
  8. 8.
    J. S. Provan, The complexity of reliability computations in planar and acyclic graphs, SIAM J. Comput., 15 (1986), pp. 694–702.Google Scholar
  9. 9.
    K. Takamizawa, T. Nishizeki, and N. Saito, Linear-time computability of combinatorial problems on series-parallel graphs, J. Assoc. Comput. Mach., 29 (1982), pp. 623–641.Google Scholar
  10. 10.
    J. Valdes, R. Tarjan, and E. Lawler, The recognition of series parallel digraphs, SIAM J. Comput., 11 (1982), pp. 298–313.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Valeska Naumann
    • 1
  1. 1.Fachbereich MathematikTU BerlinBerlin

Personalised recommendations