Advertisement

L(2,1,1)-Labeling Is NP-Complete for Trees

  • Petr A. Golovach
  • Bernard Lidický
  • Daniël Paulusma
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6108)

Abstract

An L(p 1,p 2,p 3)-labeling of a graph G with span λ is a mapping f that assigns each vertex u of G an integer label 0 ≤ f(u) ≤ λ such that |f(u) − f(v)| ≥ p i whenever vertices u and v are of distance i for i ∈ {1,2,3}. We show that testing whether a given graph has an L(2,1,1)-labeling with some given span λ is NP-complete even for the class of trees.

Keywords

Polynomial Time Chromatic Number Vertex Cover Conjunctive Normal Form Truth Assignment 
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. 1.
    Bertossi, A.A., Pinotti, M.C., Rizzi, R.: Channel assignment on strongly-simplicial graphs. In: 17th International Symposium on Parallel and Distributed Processing, p. 222. IEEE Computer Society, Washington (2003)Google Scholar
  2. 2.
    Calamoneri, T.: The L(h,k)-labelling problem: a survey and annotated bibliography. Comput. J. 49, 585–608 (2006)CrossRefGoogle Scholar
  3. 3.
    Chang, G.J., Ke, W.T., Kuo, D., Liu, D.F., Yeh, R.K.: On L(d,1)-labelings of graphs. Discrete Math. 220, 57–66 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chang, G.J., Kuo, D.: The L(2,1)-labeling problem on graphs. SIAM J. Discrete Math. 9, 309–316 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Courcelle, B.: The monadic second-order logic of graphs. I: recognizable sets of finite graphs. Inform. and Comput. 85, 12–75 (1990)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Fiala, J., Golovach, P.A., Kratochvíl, J.: Elegant distance constrained labelings of trees. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 58–67. Springer, Heidelberg (2004)Google Scholar
  7. 7.
    Fiala, J., Golovach, P.A., Kratochvíl, J.: Distance constrained labelings of graphs of bounded treewidth. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 360–372. Springer, Heidelberg (2005)Google Scholar
  8. 8.
    Fiala, J., Golovach, P.A., Kratochvíl, J.: Computational complexity of the distance constrained labeling problem for trees. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 294–305. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    Fiala, J., Golovach, P.A., Kratochvíl, J.: Parameterized complexity of coloring problems: treewidth versus vertex cover. In: TAMC 2009. LNCS, vol. 5532, pp. 221–230. Springer, Heidelberg (2009)Google Scholar
  10. 10.
    Fiala, J., Kloks, T., Kratochvíl, J.: Fixed-parameter complexity of lambda-labelings. Discrete Appl. Math. 113, 59–72 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fiala, J., Kratochvíl, J.: Partial covers of graphs. Discuss. Math. Graph Theory 22, 89–99 (2002)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Fiala, J., Kratochvíl, J., Proskurowski, A.: Distance constrained labeling of precolored trees. In: Restivo, A., Ronchi Della Rocca, S., Roversi, L. (eds.) ICTCS 2001. LNCS, vol. 2202, pp. 285–292. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Garey, M.R., Johnson, D.R.: Computers and Intractability. Freeman, New York (1979)zbMATHGoogle Scholar
  14. 14.
    Golovach, P.A.: Systems of pairs of q-distant representatives, and graph colorings. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov (POMI) 293, 5–25 (2002)Google Scholar
  15. 15.
    Golovach, P.A.: Distance-constrained labelings of trees. Vestn. Syktyvkar. Univ. Ser. 1 Mat. Mekh. Inform. 6, 67–78 (2006)MathSciNetGoogle Scholar
  16. 16.
    Liu, D., Zhu, Z.: Circular distance two labellings and circular chromatic numbers. Ars Combin. 69, 177–183 (2003)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Yeh, R.K.: A survey on labeling graphs with a condition at distance two. Discrete Math. 306, 1217–1231 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Zhou, X., Kanari, Y., Nishizeki, T.: Generalized vertex-coloring of partial k-trees. IEICE Trans. Fundamentals of Electronics, Communication and Computer Sciences E83-A, 671–678 (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Petr A. Golovach
    • 1
  • Bernard Lidický
    • 2
  • Daniël Paulusma
    • 1
  1. 1.Department of Computer ScienceUniversity of Durham, Science LaboratoriesDurhamEngland
  2. 2.Faculty of Mathematics and Physics, DIMATIA and Institute for Theoretical Computer Science (ITI)Charles UniversityPragueCzech Republic

Personalised recommendations