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Mathematical systems theory

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

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

  • Satoru Miyano
Article

Abstract

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.

Keywords

Computational Mathematic Complexity Feature Graph Property Hereditary Property Subgraph Problem 
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.

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References

  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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