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International Conference on Theory and Applications of Models of Computation

TAMC 2010: Theory and Applications of Models of Computation pp 211–221Cite as

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

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,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.

Supported by EPSRC (EP/GO43434/1) and the Royal Society ((JP090172).

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References

  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. Calamoneri, T.: The L(h,k)-labelling problem: a survey and annotated bibliography. Comput. J. 49, 585–608 (2006)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chang, G.J., Kuo, D.: The L(2,1)-labeling problem on graphs. SIAM J. Discrete Math. 9, 309–316 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Courcelle, B.: The monadic second-order logic of graphs. I: recognizable sets of finite graphs. Inform. and Comput. 85, 12–75 (1990)

    MATH  MathSciNet  Google Scholar 

  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. 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. 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)

    Chapter  Google Scholar 

  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. Fiala, J., Kloks, T., Kratochvíl, J.: Fixed-parameter complexity of lambda-labelings. Discrete Appl. Math. 113, 59–72 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fiala, J., Kratochvíl, J.: Partial covers of graphs. Discuss. Math. Graph Theory 22, 89–99 (2002)

    MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  13. Garey, M.R., Johnson, D.R.: Computers and Intractability. Freeman, New York (1979)

    MATH  Google Scholar 

  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. Golovach, P.A.: Distance-constrained labelings of trees. Vestn. Syktyvkar. Univ. Ser. 1 Mat. Mekh. Inform. 6, 67–78 (2006)

    MathSciNet  Google Scholar 

  16. Liu, D., Zhu, Z.: Circular distance two labellings and circular chromatic numbers. Ars Combin. 69, 177–183 (2003)

    MATH  MathSciNet  Google Scholar 

  17. Yeh, R.K.: A survey on labeling graphs with a condition at distance two. Discrete Math. 306, 1217–1231 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  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 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Durham, Science Laboratories, South Road, Durham, DH1 3LE, England

    Petr A. Golovach & Daniël Paulusma

  2. Faculty of Mathematics and Physics, DIMATIA and Institute for Theoretical Computer Science (ITI), Charles University, Malostranské nám. 2/25, 118 00, Prague, Czech Republic

    Bernard Lidický

Authors
  1. Petr A. Golovach
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  2. Bernard Lidický
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  3. Daniël Paulusma
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Editor information

Editors and Affiliations

  1. Charles University, Malostranske nam 25, 11800, Praha 1, Czech Republic

    Jan Kratochvíl , Jiří Fiala  & Petr Kolman ,  & 

  2. State Key Lab. of Computer Science, Institute of Software,, Chinese Academy of Sciences, 100190, Beijing, P.R. China

    Angsheng Li

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Golovach, P.A., Lidický, B., Paulusma, D. (2010). L(2,1,1)-Labeling Is NP-Complete for Trees. In: Kratochvíl, J., Li, A., Fiala, J., Kolman, P. (eds) Theory and Applications of Models of Computation. TAMC 2010. Lecture Notes in Computer Science, vol 6108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13562-0_20

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  • DOI: https://doi.org/10.1007/978-3-642-13562-0_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13561-3

  • Online ISBN: 978-3-642-13562-0

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