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Computational Complexity of the Distance Constrained Labeling Problem for Trees (Extended Abstract)

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Automata, Languages and Programming (ICALP 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5125))

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Abstract

An L(p,q)-labeling of a graph is a labeling of its vertices by nonnegative integers such that the labels of adjacent vertices differ by at least p and the labels of vertices at distance 2 differ by at least q. The span of such a labeling is the maximum label used. Distance constrained labelings are an important graph theoretical approach to the Frequency Assignment Problem applied in mobile and wireless networks.

In this paper we show that determining the minimum span of an L(p,q)-labeling of a tree is NP-hard whenever q is not a divisor of p. This demonstrates significant difference in computational complexity of this problem for q = 1 and q > 1. In addition, we give a sufficient and necessary condition for the existence of an H(p,q)-labeling of a tree in the case when the metric on the label space is determined by a strongly vertex transitive graph H. This generalizes the problem of distance constrained labeling in cyclic metric, that was known to be solvable in polynomial time for trees.

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References

  1. Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12(2), 308–340 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  2. Calamoneri, T.: The L(h,k)-labeling problem: A survey and annotated bibliography. Computer Journal 49(5), 585–608 (2006)

    Article  Google Scholar 

  3. Chang, G.J., Ke, W.-T., Liu, D.D.-F., Yeh, R.K.: On L(d,1)-labelings of graphs. Discrete Mathematics 220(1–3), 57–66 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  5. 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)

    Chapter  Google Scholar 

  6. Fiala, J., Golovach, P. A., and Kratochvíl, J.: Distance constrained labelings of trees. Tech. Rep. ITI Series 2008–369, Charles University (2007)

    Google Scholar 

  7. Fiala, J., Kratochvíl, J.: Partial covers of graphs. Discussiones Mathematicae Graph Theory 22, 89–99 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fiala, J., Kratochvíl, J., Kloks, T.: Fixed-parameter complexity of λ-labelings. Discrete Applied Mathematics 113(1), 59–72 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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 

  10. Fiala, J., Kratochvíl, J., Proskurowski, A.: Systems of distant representatives. Discrete Applied Mathematics 145(2), 306–316 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Griggs, J.R., Yeh, R.K.: Labelling graphs with a condition at distance 2. SIAM Journal on Discrete Mathematics 5(4), 586–595 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Harary, F.: Graph theory. Addison-Wesley Series in Mathematics IX. Addison-Wesley, Reading (1969)

    MATH  Google Scholar 

  13. Havet, F., Reed, B., Sereni, J.-S.: L(2,1)-labelling of graphs. In: Huang, S.-T. (ed.) Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, San Francisco, California, USA, January 20-22, 2008, pp. 621–630. SIAM, Philadelphia (2008)

    Google Scholar 

  14. Král, D.: Mixed hypergraphs and other coloring problems. Discrete Mathematics 307(7-8), 923–938 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Leese, R.A., Noble, S.D.: Cyclic labellings with constraints at two distances. Electronic Journal of Combinatorics 11(1) (2004)

    Google Scholar 

  16. Liu, D.D.-F., Zhu, X.: Circulant distant two labeling and circular chromatic number. Ars Combinatoria 69, 177–183 (2003)

    MathSciNet  MATH  Google Scholar 

  17. Matoušek, J., Nešetřil, J.: Invitation to Discrete Mathematics. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  18. Roberts, F.S.: T-colorings of graphs: recent results and open problems. Discrete Mathematics 93(2–3), 229–245 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yeh, R.K.: A survey on labeling graphs with a condition at distance two. Discrete Mathematics 306(12), 1217–1231 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute for Theoretical Computer Science and Department of Applied Mathematics, Charles University, Prague, Czech Republic

    Jiří Fiala & Jan Kratochvíl

  2. Institutt for informatikk, Universitetet i Bergen, Norway

    Petr A. Golovach

Authors
  1. Jiří Fiala
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  2. Petr A. Golovach
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  3. Jan Kratochvíl
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Editor information

Editors and Affiliations

  1. School of Computer Science, Reykjavik University, Kringlan 1, 103, Reykjavík, Iceland

    Luca Aceto

  2. Department of Computer Science, IT-Parken, University of Aarhus, Åbogade 34, 8200, ÅrhusN, Denmark

    Ivan Damgård

  3. Department of Computer Science, University of Liverpool, Ashton Building, L69 3BX, Liverpool, UK

    Leslie Ann Goldberg

  4. School of Computer Science, Reykjavik University, Kringlan 1, 103, Reykjavik, Iceland

    Magnús M. Halldórsson

  5. Department of Computer Science, Reykjavik University, Kringlan 1, 103, Reykjavík, Iceland

    Anna Ingólfsdóttir

  6. Université de Bordeaux-1, LaBRI, 351, Cours de la Libération, 33405, Talence cedex, France

    Igor Walukiewicz

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© 2008 Springer-Verlag Berlin Heidelberg

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Fiala, J., Golovach, P.A., Kratochvíl, J. (2008). Computational Complexity of the Distance Constrained Labeling Problem for Trees (Extended Abstract). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70575-8_25

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  • DOI: https://doi.org/10.1007/978-3-540-70575-8_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-70574-1

  • Online ISBN: 978-3-540-70575-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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