Parallel complexity of lexicographically first problems for tree-structured graphs

  • Bogdan Chlebus
  • Krzysztof Diks
  • Wojciech Rytter
  • Tomasz Szymacha
Part of the Lecture Notes in Computer Science book series (LNCS, volume 379)


We study several P-complete graph problems and show that they are in NC if the input graphs are restricted to be tree-structured. These graphs are also known as partial k-trees, decomposable graphs or graphs of bounded tree width, and include outerplanar graphs, series-parallel graphs and Halin graphs. The particular problems investigated herein include the lexicographically first (l.f.) depth-first search tree and the l.f. maximal independent set. It is also shown that if a tree of faces of an outerplanar graph is given, then its dfs tree can be found in O(log2n) time using O(n/log2n) processors.


Decomposition Tree Outerplanar Graph Halin Graph Optimal Parallel Algorithm Decomposable Graph 
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. [AM 78]
    R. Anderson,E.W. Meyr, Parallelism and the maximal path problem. Inf.Procc.Letters 24 (1978) 121–126.Google Scholar
  2. [ALS 88]
    S.Arnborg,J.Lagergren,D.Seese, Problems easy for tree-decomposable graphs. ICALP (1988) 38–51.Google Scholar
  3. [Arn 85]
    S. Arnborg, Efficient Algorithms for combinatorial problems on graphs with bounded decomposability — a survey. BIT 25 (1985), 2–33.Google Scholar
  4. [ACP 87]
    S. Arnborg,D.G. Corneil,A. Proskurowski, Complexity of finding embeddings in a k-tree. SIAM J.Alg. and Discr. Methods 8 (1987), 277–284.Google Scholar
  5. [BLW 87]
    W.M. Bern,E.L. Lawler and A.L. Wong, Linear time computation of optimal subgraphs of decomposable graphs. J. of Algorithms 8(1987) 216–235.Google Scholar
  6. [Bo 86]
    H.L. Bodlaender, Classes of graphs with bounded tree-width, Preprint, Dec 1986.Google Scholar
  7. [Bo 87a]
    H.L. Bodlaender, Dynamic programming on graphs with bounded tree width, ICALP (1988).Google Scholar
  8. [Bo 87b]
    H.L. Bodlaender, Polynomial algorithms for chromatic index and graph isomorphism on patrial k-trees. Technical Report RUU-CS-87-17 Dept.of Comp.Science, Univ. of Utrecht, Utrecht, 1987.Google Scholar
  9. [Bo 88]
    H.L.Bodlaender, NC-algorithms for graphs with small tree width. Technical Report RUU-CS-88-4.Google Scholar
  10. [CDR 88]
    B.S.Chlebus,K.Diks,T.Radzik, Testing isomorphism of outerplanar graphs in parallel. Proc. MFCS (1988).Google Scholar
  11. [Coo 85]
    S.A. Cook, A taxonomy of problems with fast parallel algorithms. Inf.Control 64(1985) 2–22.Google Scholar
  12. [DHR 89]
    K.Diks, T.Hagerup, W.Rytter, Optimal parallel algorithms for the recognition and coloring outerplanar graphs. This proceedings.Google Scholar
  13. [GR 88]
    A.Gibbons,W.Rytter, Efficient parallel algorithms. Cambridge University Press (1988).Google Scholar
  14. [GR 87]
    A.Gibbons, W.Rytter, Fast parallel edge coloring of tree structured graphs. FCT (1987).Google Scholar
  15. [Miy 87]
    S.Miyano, The lexicographically first maximal subgraph problems: P-completeness and NC algorithms. ICALP'87.Google Scholar
  16. [Rei 85]
    J. Reif, Depth first search is inherently sequential. Inf.Proc.Lettres 20(1985) 229–234.Google Scholar
  17. [RS 86]
    N. Robertson and P. Seymour, Graph minors II. Algoritmic aspects of tree width. J. of Algorithms, 7:309–322, 1986.Google Scholar
  18. [RSz 89]
    W.Rytter,T.Szymacha, Parallel algorithms for a clas of graphs defined recursively. Accepted for Inf. Proc. Letters.Google Scholar
  19. [Ryt 85]
    W.Rytter, The complexity of two way pushdown automata and recursive programs. Combinatorial algorithms on words (ed.A.Apostolico, Z.Galil) Springer-Verag (1985).Google Scholar
  20. [Sli 82]
    A. Slisenko, Context-free grammars as a tool for describing polynomial time subclasses of hard problems. Inf.Proc.Letters 14:2 (1982) 52–57.Google Scholar
  21. [Smi 86]
    J. Smith, Parallel algorithms for depth first search. I. Planar graphs. SIAM J.Comp. 15 (1986) 814–830.Google Scholar
  22. [Sys 83]
    M.M.Syslo, NP-complete problems on some tree-structured graphs: a review. In M.Nagl and J.Perl, editors, Proc.WG'83 International Workshop on Graph Therotecic Concepts in Computer Science, 342–353, (1983).Google Scholar
  23. [TV 84]
    R.Tarjan, U.Vishkin, Finding biconnected components and computing tree functions in logarithmic parallel time. 25th IEEE FOCS (1984) 12–20.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Bogdan Chlebus
    • 1
  • Krzysztof Diks
    • 1
  • Wojciech Rytter
    • 1
  • Tomasz Szymacha
    • 1
  1. 1.Instytut InformatykiUniwersytet WarszawskiWarszawaPoland

Personalised recommendations