Mathematical systems theory

, Volume 22, Issue 1, pp 47–73 | Cite as

The lexicographically first maximal subgraph problems:P-completeness andNC algorithms

  • Satoru Miyano


The lexicographically first maximal (lfm) subgraph problem for a property π is to compute the lfm vertex set whose induced subgraph satisfies π. The main contribution of this paper is theP-completeness of the lfm subgraph problem for any nontrivial hereditary property. We also observe that most of the lfm subgraph problems are stillP-complete even if the instances are restricted to graphs with degree 3. However, some exceptions are found. For example, it is shown that the lfm 4-cycle free subgraph problem is inNC2 for graphs with degree 3 but turns out to beP-complete for graphs with degree 4. Further, we analyze the complexity of the lfmedge-induced subgraph problem for some graph properties and show that it has a different complexity feature.


Computational Mathematic Complexity Feature Graph Property Hereditary Property Subgraph Problem 
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  1. [1]
    R. Anderson and E. W. Mayr, Parallelism and greedy algorithms, STAN-CS-84-1003, April 1984.Google Scholar
  2. [2]
    R. Anderson and E. W. Mayr, Parallelism and the maximal path problem,Inform. Process. Lett.,24 (1987), 121–126.Google Scholar
  3. [3]
    T. Asano and T. Hirata, Edge-deletion and edge-contraction problems,Proc. 14th ACM STOC, 1982, pp. 245–254.Google Scholar
  4. [4]
    J. Avenhaus and K. Madlener, The Nielsen reduction andP-complete problems in free groups,Theoret. Comput. Sci.,32 (1984), 61–74.Google Scholar
  5. [5]
    K. S. Booth and G. S. Leuker, Testing for the consecutive ones property, interval graphs, and graph planarity usingPQ-tree algorithms,J. Comput. System Sci.,13 (1976), 335–379.Google Scholar
  6. [6]
    G. Chartrand and F. Harary, Planar permutation graphs,Ann. Inst. Henri Poincaré,4 (1967), 433–438.Google Scholar
  7. [7]
    S. A. Cook, An observation on time-storage trade off,J. Comput. System Sci.,9 (1974), 308–316.Google Scholar
  8. [8]
    S. A. Cook, A taxonomy of problems with fast parallel algorithms,Inform. and Control,64 (1985), 2–22.Google Scholar
  9. [9]
    D. Dobkin, R. J. Lipton, and S. Reiss, Linear programming is log-space hard forP, Inform. Process. Lett.,8 (1979), 96–97.Google Scholar
  10. [10]
    M. T. Garey and D. S. Johnson, The rectilinear Steiner tree problem isNP-complete,SIAM J. Appl. Math.,32 (1977), 826–834.Google Scholar
  11. [11]
    L. M. Goldschlager, The monotone and planar circuit value problems are log space complete forP, SIGACT News,9 (1977), 25–29.Google Scholar
  12. [12]
    L. M. Goldschlager, R. E. Shaw, and J. Staples, The maximum flow problem is log-space complete forP, Theoret. Comput. Sci.,21 (1982), 105–111.Google Scholar
  13. [13]
    F. Harary,Graph Theory, Addison-Wesley, Reading, MA, 1970.Google Scholar
  14. [14]
    J. E. Hopcroft and R. E. Tarjan, Efficient planarity testing,J. Assoc. Comput. Mach.,21 (1974), 549–568.Google Scholar
  15. [15]
    D. B. Johnson and S. M. Venkatesan, Parallel algorithm for minimum cuts and maximum flows in planar networks,Proc. 22nd IEEE FOCS, 1982, pp. 244–254.Google Scholar
  16. [16]
    N. D. Jones and W. T. Laaser, Complete problems for deterministic polynomial time,Theoret. Comput. Sci.,3 (1977), 105–117.Google Scholar
  17. [17]
    R. M. Karp, E. Upfal, and A. Wigderson, Constructing a perfect matching is in randomNC, Proc. 17th ACM STOC, 1985, pp. 22–32.Google Scholar
  18. [18]
    R. M. Karp, E. Upfal, and A. Wigderson, Are search and decision problems computationally equivalent?,Proc. 17th ACM STOC, 1985, pp. 464–475.Google Scholar
  19. [19]
    R. M. Karp, E. Upfal, and A. Wigderson, The complexity of parallel computation on matroids,Proc. 26th IEEE FOCS, 1985, pp. 229–234.Google Scholar
  20. [20]
    R. E. Ladner, The circuit value problem is log-space complete forP, SIGACT News,7 (1975), 18–21.Google Scholar
  21. [21]
    J. M. Lewis and M. Yannakakis, The node deletion problems for hereditary properties isNP-complete,J. Comput. System Sci.,20 (1980), 219–230.Google Scholar
  22. [22]
    M. Luby, A simple parallel algorithm for the maximal independent set problem,Proc. 17th ACM STOC, 1985, pp. 1–10.Google Scholar
  23. [23]
    C. U. Martel, Lower bounds on parallel algorithms for finding the first maximal independent set,Inform. Process. Lett.,22 (1986), 81–85.Google Scholar
  24. [24]
    S. L. Mitchell, Linear algorithms recognize outerplanar and maximal outerplanar graphs,Inform. Process. Lett.,9 (1979), 229–232.Google Scholar
  25. [25]
    S. Miyano, A parallelizable lexicographically first maximal edge-induced subgraph problem,Inform. Process. Lett.,27 (1988), 75–78.Google Scholar
  26. [26]
    S. Miyano, Δ2p-complete lexicographically first maximal subgraph problems,Proc. 13th MFCS, Lecture Notes in Computer Science, Vol. 324, Springer-Verlag, Berlin, 1988, pp. 454–462.Google Scholar
  27. [27]
    S. Miyano, Parallel complexity andP-complete problems,Proc. FGCS '88, 1988, pp. 532–541.Google Scholar
  28. [28]
    N. Pippenger, On simultaneous resource bounds,Proc. 20th IEEE FOCS, 1979, pp. 307–311.Google Scholar
  29. [29]
    J. H. Reif, Depth-first search is inherently sequential,Inform. Process. Lett.,20 (1985), 229–234.Google Scholar
  30. [30]
    W. L. Ruzzo, On uniform circuit complexity,J. Comput. System Sci.,22 (1981), 365–383.Google Scholar
  31. [31]
    E. Speckenmeyer, Untersuchungen zum Feedback Vertex Set Problem in ungerichteten Graphen, Ph.D. Thesis, Universität-GH-Paderborn, 1983.Google Scholar
  32. [32]
    T. Watanabe, T. Ae, and A. Nakamura, On the removal of forbidden graphs by edge-deletion or by edge-contraction,Discrete Appl. Math.,3 (1981), 151–153.Google Scholar
  33. [33]
    M. Wiegers, Recognizing outerplanar graphs in linear time,Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, Vol. 246, Springer-Verlag, Berlin, 1986, pp. 165–176.Google Scholar
  34. [34]
    M. Yannakakis, Node-deletion problems on bipartite graphs,SIAM J. Comput.,10 (1981), 310–327.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1989

Authors and Affiliations

  • Satoru Miyano
    • 1
  1. 1.Research Institute of Fundamental Information ScienceFukuokaJapan

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