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Exact Algorithms for L(2,1)-Labeling of Graphs

  • Jan Kratochvíl
  • Dieter Kratsch
  • Mathieu Liedloff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)

Abstract

The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph G = (V,E) into an interval of integers [0..k] is an L(2,1)-labeling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbor are mapped onto distinct integers. It is known that for any fixed k ≥ 4, deciding the existence of such a labeling is an NP-complete problem. We present exact exponential time algorithms that are faster than the naive O((k + 1) n ) algorithm that would try all possible mappings. The improvement is best seen in the first NP-complete case of k = 4 – here the running time of our algorithm is O(1.3161 n ).

Keywords

Dynamic Programming Exact Algorithm Adjacent Vertex Common Neighbor Label Vertex 
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.

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References

  1. 1.
    Bodlaender, H.L., Kloks, T., Tan, R.B., van Leeuwen, J.: Approximations for lambda-Colorings of Graphs. Computer Journal 47, 193–204 (2004)zbMATHCrossRefGoogle Scholar
  2. 2.
    Chang, G.J., Kuo, D.: The L(2,1)-labeling problem on graphs. SIAM Journal of Discrete Mathematics 9, 309–316 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Fiala, J., Golovach, P., 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
  4. 4.
    Fiala, J., Kloks, T., Kratochvíl, J.: Fixed-parameter complexity of λ-labelings. Discrete Applied Mathematics 113, 59–72 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Fiala, J., Kratochvíl, J.: Complexity of partial covers of graphs. In: Eades, P., Takaoka, T. (eds.) ISAAC 2001. LNCS, vol. 2223, pp. 537–549. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  6. 6.
    Fiala, J., Kratochvíl, J.: Partial covers of graphs. Mathematicae Graph Theory 22, 89–99 (2002)zbMATHGoogle Scholar
  7. 7.
    Fiala, J., Kratochvíl, J., Pór, A.: On the computational complexity of partial covers of theta graphs. Electronic Notes in Discrete Mathematics 19, 79–85 (2005)CrossRefGoogle Scholar
  8. 8.
    Fomin, F., Grandoni, F., Kratsch, D.: Measure and conquer: Domination - A case study. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 192–203. Springer, Heidelberg (2005)Google Scholar
  9. 9.
    Fomin, F., Heggernes, P., Kratsch, D.: Exact algorithms for graph homomorphisms. In: Liśkiewicz, M., Reischuk, R. (eds.) FCT 2005. LNCS, vol. 3623, pp. 161–171. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Gonçalves, D.: On the L(p,1)-labelling of graphs. In: DMTCS Proceedings, vol. AE, pp. 81–86Google Scholar
  11. 11.
    Griggs, J.R., Yeh, R.K.: Labelling graphs with a condition at distance 2. SIAM Journal of Discrete Mathematics 5, 586–595 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hell, P., Nešetřil, J.: On the complexity of H-colouring. Journal of Combinatorial Theory Series B 48, 92–110 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Leese, R.A., Noble, S.D.: Cyclic labellings with constraints at two distances. Electronic Journal of Combinatorics, Research paper 16, 11, (2004)Google Scholar
  14. 14.
    Liu, D., Zhu, X.: Circular Distance Two Labelings and Circular Chromatic Numbers. Ars Combinatoria 69, 177–183 (2003)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Roberts, F.S.: Private communication to J. GriggsGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Jan Kratochvíl
    • 1
  • Dieter Kratsch
    • 2
  • Mathieu Liedloff
    • 2
  1. 1.Department of Applied Mathematics, and Institute for Theoretical Computer Science, Charles University, Malostranské nám. 25, 118 00 Praha 1Czech Republic
  2. 2.Laboratoire d’Informatique Théorique et Appliquée, Université Paul Verlaine - Metz, 57045 Metz Cedex 01France

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