NP-Completeness of Some Tree-Clustering Problems

  • Falk Schreiber
  • Konstantinos Skodinis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1547)

Abstract

A graph is a tree of paths (cycles), if its vertex set can be partitioned into clusters, such that each cluster induces a simple path (cycle), and the clusters form a tree. Our main result states that the problem whether or not a given graph is a tree of paths (cycles) is NP-complete. Moreover, if the length of the paths (cycles) is bounded by a constant, the problem is in P.

References

  1. [1]
    S. Arnborg, D. Corneil, and A. Proskurowski. Complexity of finding embeddings in a k-tree. SIAM J. Alg. Disc. Meth., 8:277–384, 1987.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    F. T. Boesch and J. F. Gimpel. Covering the points of a digraph with point-disjoint paths and its application to code optimization. J. Assoc. Comput. Mach., 24(2):192–198, 1977.MATHMathSciNetGoogle Scholar
  3. [3]
    F. J. Brandenburg. Graph clustering I: Cycles of cliques. In G. Di Battista, editor, Proc. of Graph Drawing, volume 1353 of Lect. Notes in Comput. Sci., pages 158–168. Springer-Verlag, New York/Berlin, 1997.CrossRefGoogle Scholar
  4. [4]
    E. Cohen and M. Tarsi. NP-completeness of graph decomposition problems. J. Complexity, 7:200–212, 1991.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    D. Dor and M. Tarsi.Graph decomposition is NP-complete: A complete proof of holyer’s conjecture. SIAM J. Comput., 26(4):1166–1187, 1997.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    P. Eades and Q.-W. Feng. Multilevel visualization of clustered graphs. In S. North, editor, Proc. of Graph Drawing, volume 1190 of Lect. Notes in Comput. Sci., pages 101–112. Springer-Verlag, New York/Berlin, 1996.Google Scholar
  7. [7]
    Q.-W. Feng, R. F. Cohen, and P. Eades. How to draw a planar clustered graph. In D.-Z. Du, editor, Computing and Combinatorics, volume 959 of Lect. Notes in Comput. Sci., pages 21–30. Springer-Verlag, New York/Berlin, 1995.CrossRefGoogle Scholar
  8. [8]
    T. Fruchterman and E. Reingold. Graph drawing by force-directed placement. Software-Practice and Experience, 21(11):1129–1164, 1991.CrossRefGoogle Scholar
  9. [9]
    M. R. Garey and D. S. Johnson. Computers and Intractability;A guide to the theory of NP-completeness. W. H. Freeman, 1979.Google Scholar
  10. [10]
    M. Himsolt. The graphlet system. In S. North, editor, Proc. of Graph Drawing, volume 1190 of Lect. Notes in Comput. Sci., pages 233–240. Springer-Verlag, New York/Berlin, 1996.Google Scholar
  11. [11]
    I. Holyer. The NP-completeness of some edge-partition problems. SIAM J. Comput., 10(4):713–717, 1981.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    J. Kratochvil, M. Goljan, and P. Kucaera. String graphs. Technical report, Academia Prague, 1986.Google Scholar
  13. [13]
    C. Papadimitriou. Computational Complexity. Addison-Wesley Publishing Company, 1994.Google Scholar
  14. [14]
    T. Roxborough and A. Sen. Graph clustering using multiway ratio cut. In G. Di-Battista, editor, Proc. of Graph Drawing, volume 1353 of Lect. Notes in Comput. Sci., pages 291–296. Springer-Verlag, New York/Berlin, 1997.CrossRefGoogle Scholar
  15. [15]
    K. Sugiyama, S. Tagawa, and M. Toda. Methods for visual understanding of hierarchical systems. IEEE Transactions on Systems, Man and Cybernetics, 11:109–125, 1981.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Falk Schreiber
    • 1
  • Konstantinos Skodinis
    • 1
  1. 1.University of PassauPassauGermany

Personalised recommendations