Advertisement

Min Cut is NP-complete for edge weighted trees

  • B. Monien
  • I. H. Sudborough
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 226)

Abstract

We show that the Min Cut Linear Arrangement Problem is NP-complete for trees with polynomial size weights and derive from this the NP-completeness of Min Cut for planar graphs with maximum vertex degree 3. This is used to show the NP-completeness of Search Number, Vertex Separation, Progressive Black/White Pebble Demand, and Topological Bandwidth for planar graphs with maximum vertex degree 3.

Keywords

Planar Graph Center Vertex Pebble Game Maximum Vertex Degree Vertex Separation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [C]
    F. R. K. Chung, "On the Cutwidth and the Topological Bandwidth of a Tree", SIAM J. Alg. Discrete Meth. (1985).Google Scholar
  2. [CMSTZZZ]
    M.-J. Chung, F. Makedon, I. H. Sudborough and J. Turner, "Polynomial Algorithms for the Min-Cut Linear Arrangement Problem on Degree Restricted Trees", SIAM J. Computing 14,1 (1985), pp. 158–177.Google Scholar
  3. [CS]
    S. A. Cook and R. Sethi, "Storage Requirements for Deterministic Polynomial Time Recognizable Languages", J. Comput. Syst. Sci. 13 (1976), pp. 25–37.Google Scholar
  4. [ESTZZZ]
    J.A. Ellis, I.H. Sudborough, and J.S. Turner, "Graph Separation and Search Number", Proc. 1983 Allerton Conf. on Communication, Control, and Computing.Google Scholar
  5. [GJ]
    Garey, M. R and Johnson, D. S, "Computers and Intractability: A Guide to the Theory of NP-Completeness", W. H. Freeman and Company, San Francisco (1979)Google Scholar
  6. [GGJK]
    Garey, M. R, Graham, R. L, Johnson, D. S and Knuth, D. E, "Complexity Results for Bandwidth Minimization", SIAM J. Applied Math. 34 (1978), pp. 477–495.Google Scholar
  7. [GLTZZZ]
    Gilbert, J. R, Lengauer, T and Tarjan, R. E, "The Pebbling Problem is Complete in Polynomial Space", SIAM Journal on Computing 9,3 1980 pp. 513–524Google Scholar
  8. [HPV]
    Hopcroft, J. E, Paul, W and Valiant, L, "On Time Versus Space", Journal ACM, 24, (1977), pp. 332–337Google Scholar
  9. [KP]
    M. Kirousis and C. H. Papadimitriou, "Searching and Pebbling", Technical Report, National Technical University, Athens, Greece (1983).Google Scholar
  10. [LaP]
    S. LaPaugh, "Recontamination does not Help to Search a Graph", Technical Report, Electrical Engineering and Computer Science Department, Princeton University (1983)Google Scholar
  11. [L]
    T. Lengauer, "Black-White Pebbles and Graph Separation", SIAM J. Algebraic and Discrete Methods, 1982.Google Scholar
  12. [LTZZZ]
    R. J. Lipton and R. E. Tarjan, "A Separator Theorem for Planar Graphs", SIAM J. Appl. Math. 36,2 (1979), pp. 177–189Google Scholar
  13. [MadH]
    F. Meyer auf der Heide, "A Comparison of Two Variations of a Pebble Game on Graphs", Theoretical Computer Science 13 (1981), pp. 315–322.Google Scholar
  14. [MS]
    F. Makedon and I. H. Sudborough, "Minimizing Width in Linear Layouts", Proc. 10th ICALP, vol. 154, Lecture Notes in Computer Science, Springer Verlag (1983), pp. 478–490.Google Scholar
  15. [MPS]
    F. Makedon, C. H. Papadimitriou and I. H. Sudborough, "Topological Bandwidth", SIAM J. Alg. Discrete Meth. 6,3 (1985), pp. 418–444.Google Scholar
  16. [MHGKP]
    N. Megiddo, S. L. Hakimi, M. R. Garey, D. S. Johnson and C. H. Papadimitriou, "The Complexity of Searching a Graph (Preliminary Version)", Proc. IEEE Foundations of Computer Science Symp. (1981), pp. 376–385Google Scholar
  17. [Mo]
    B. Monien, personal communication.Google Scholar
  18. [P]
    T. D. Parsons, "Pursuit-Evasion in a Graph", in Theory and Application of Graphs, Y. Alavi and D. R. Lick, (eds) Springer-Verlag, Berlin, 1976, pp. 426–441Google Scholar
  19. [P2]
    T. D. Parsons, "The Search Number of a Connected Graph", Proc. 9th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, Utilitas Mathematica Publishing, Winnipeg, (1978), pp. 549–554Google Scholar
  20. [W]
    R. Wilber, "White Pebbles Help", Proc. 17th Annual ACM Symposium on Theory of Computing, 1985, pp. 103–112.Google Scholar
  21. [Y]
    M. Yannakais, "A Polynomial Algorithm for the Min Cut Linear Arrangement of Trees", J. ACM 32,4 (1985), pp. 950–959.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • B. Monien
    • 1
  • I. H. Sudborough
    • 2
  1. 1.Fachbereich Mathematik und InformatikUniversitat PaderbornPaderbornBRD
  2. 2.Computer Science DepartmentUniversity of Texas-DallasRichardsonUSA

Personalised recommendations