Bounded Degree Closest k-Tree Power Is NP-Complete

  • Michael Dom
  • Jiong Guo
  • Rolf Niedermeier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3595)


An undirected graph G=(V,E) is the k-power of an undirected tree T=(V,E′) if (u,v)∈ E iff u and v are connected by a path of length at most k in T. The tree T is called the tree root of G. Tree powers can be recognized in polynomial time. The thus naturally arising question is whether a graph G can be modified by adding or deleting a specified number of edges such that G becomes a tree power. This problem becomes NP-complete for k≥ 2. Strengthening this result, we answer the main open question of Tsukiji and Chen [COCOON 2004] by showing that the problem remains NP-complete when additionally demanding that the tree roots must have bounded degree.


Vertex Cover Chordal Graph Edge Node Tree Power Induce Subgraph 
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. 1.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: a Survey. SIAM Monographs on Discrete Mathematics and Applications (1999)Google Scholar
  2. 2.
    Chen, Z.-Z., Jiang, T., Lin, G.: Computing phylogenetic roots with bounded degrees and errors. SIAM Journal on Computing 32(4), 864–879 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, Z.-Z., Tsukiji, T.: Computing bounded-degree phylogenetic roots of disconnected graphs. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 308–319. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Dom, M., Guo, J., Hüffner, F., Niedermeier, R.: Error compensation in leaf root problems. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 389–401. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Dom, M., Guo, J., Hüffner, F., Niedermeier, R.: Extending the tractability border for closest leaf powers. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 397–408. Springer, Heidelberg (2005) (to appear)CrossRefGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.J.: Some simplified NP-complete graph problems. Theoretical Computer Science 1(3), 237–267 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jiang, T., Lin, G., Xu, J.: On the closest tree kth root problem. Manuscript, Department of Computer Science, University of Waterloo (2000)Google Scholar
  8. 8.
    Kearney, P.E., Corneil, D.G.: Tree powers. Journal of Algorithms 29(1), 111–131 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Křivánek, M., Morávek, J.: NP-hard problems in hierarchical-tree clustering. Acta Informatica 23(3), 311–323 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lau, L.C., Corneil, D.G.: Recognizing powers of proper interval, split, and chordal graphs. SIAM Journal on Discrete Mathmatics 18(1), 83–102 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lin, Y.L., Skiena, S.S.: Algorithms for square roots of graphs. SIAM Journal on Discrete Mathematics 8(1), 99–118 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Motwani, R., Sudan, M.: Computing roots of graphs is hard. Discrete Applied Mathematics 54(1), 81–88 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nishimura, N., Ragde, P., Thilikos, D.M.: On graph powers for leaf-labeled trees. Journal of Algorithms 42(1), 69–108 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Tsukiji, T., Chen, Z.-Z.: Computing phylogenetic roots with bounded degrees and errors is hard. In: Chwa, K.-Y., Munro, J.I.J. (eds.) COCOON 2004. LNCS, vol. 3106, pp. 450–461. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Michael Dom
    • 1
  • Jiong Guo
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.Institut für InformatikFriedrich-Schiller-Universität JenaJenaFed. Rep. of Germany

Personalised recommendations