Fixing Improper Colorings of Graphs

  • Konstanty Junosza-Szaniawski
  • Mathieu Liedloff
  • Paweł Rzążewski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8939)

Abstract

In this paper we consider a variation of a recoloring problem, called the r-Color-Fixing. Let us have some non-proper r-coloring ϕ of a graph G. We investigate the problem of finding a proper r-coloring of G, which is “the most similar” to ϕ, i.e. the number k of vertices that have to be recolored is minimum possible. We observe that the problem is NP-complete for any r ≥ 3, but is Fixed Parameter Tractable (FPT), when parametrized by the number of allowed transformations k. We provide an \(\mathcal{O}^*(2^n)\) algorithm for the problem (for any fixed r) and a linear algorithm for graphs with bounded treewidth. Finally, we investigate the fixing number of a graph G. It is the maximum possible distance (in the number of transformations) between some non-proper coloring of G and a proper one.

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References

  1. 1.
    Ausiello, G., Escoffier, B., Monnot, J., Paschos, V.T.: Reoptimization of minimum and maximum traveling salesmans tours. J. of Discrete Algorithms 7, 453–463 (2009)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bilò, D., Böckenhauer, H.-J., Komm, D., Královic, R., Mömke, T., Seibert, S., Zych, A.: Reoptimization of the Shortest Common Superstring Problem. Algorithmica 61, 227–251 (2011)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM Journal on Computing 39, 546–563 (2009)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bodlaender, H.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25, 1305–1317 (1996)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bodlaender H., Koster A.: Combinatorial Optimization on Graphs of Bounded Treewidth. The Computer Journal (2007)Google Scholar
  6. 6.
    Bonamy, M., Bousquet, N.: Recoloring bounded treewidth graphs. Electronic Notes in Discrete Mathematics 44, 257–262 (2013)CrossRefGoogle Scholar
  7. 7.
    Bonsma, P., Cereceda, L.: Finding Paths Between Graph Colourings: PSPACE-Completeness and Superpolynomial Distances. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 738–749. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Cereceda, L., van den Heuvel, J., Johnson, M.: Mixing 3-Colourings in Bipartite Graphs. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 166–177. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Cereceda, L., Heuvel, J., van den Johnson, M.: Connectedness of the graph of vertex colourings. Discrete Mathematics 308, 166–177 (2008)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Cereceda, L., van den Heuvel, J., Johnson, M.: Finding paths between 3-colorings. Journal of Graph Theory 67, 69–82 (2011)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Diestel, R.: Graph Theory, 3rd edn. Graduate Texts in Mathematics. Springer (2005)Google Scholar
  12. 12.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer (1999)Google Scholar
  13. 13.
    Felsner, S., Huemer, C., Saumell, M.: Recoloring directed graphs. In: Proc. of XIII Encuentros de Geometría Computacional, pp. 91–97 (2009)Google Scholar
  14. 14.
    Jerrum, M.: A very simple algorithm for estimating the number of k-colorings of a low-degree graph. Random Structures & Algorithms 7, 157–165 (1995)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Kloks, T. (ed.): Treewidth. LNCS, vol. 842. Springer, Heidelberg (1994)MATHGoogle Scholar
  16. 16.
    Shachnai, H., Tamir, G., Tamir, T.: A Theory and Algorithms for Combinatorial Reoptimization. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 618–630. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  17. 17.
    Zych, A., Bilò, D.: New Reoptimization Techniques applied to Steiner Tree Problem. Electronic Notes in Discrete Mathematics 37(2–1), 387–392Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Konstanty Junosza-Szaniawski
    • 1
  • Mathieu Liedloff
    • 2
  • Paweł Rzążewski
    • 1
  1. 1.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarszawaPoland
  2. 2.INSA Centre Val de Loire, LIFOUniversité d’OrléansOrléansFrance

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