Advertisement

(H,C,K) -Coloring: Fast, Easy, and Hard Cases

  • Josep Díaz
  • Maria Serna
  • Dimitrios M. Thilikos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2136)

Abstract

We define a variant of the H-coloring problem by fixing the number of preimages of a subset C of the vertices of H, thus allowing parameterization. We provide sufficient conditions to guarantee that the problem can be solved in O(kn + f(k, H)) steps where f is a function depending only on the number k of fixed preimages and the graph H, and in O(n k+c) steps where c is a constant independent of k. Finally, we prove that whenever the non parameterized vertices induce in G a graph that is bipartite and loopless the problem is NP-complete.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [DF99]
    R. G. Downey and M. R. Fellows. Parameterized complexity. Springer-Verlag, New York, 1999.Google Scholar
  2. [DST01a]
    Josep Díaz, Maria Serna, and Dimitrios M. Thilikos. Counting list H- colorings and variants. Technical Report LSI-01-27-R, Departament de Llenguatges i Sistemes Informàtics, Universitat Politècnica de Catalunya, Barcelona, Spain, 2001.Google Scholar
  3. [DST01b]
    Josep Díaz, Maria Serna, and Dimitrios M. Thilikos. Evaluating H-colorings of graphs with bounded decomposability. In Proceedings of The 7th Annual International Computing and Combinatorics Conference, COCOON 2001, 2001.Google Scholar
  4. [FH98]
    Tomas Feder and Pavol Hell. List homomorphisms to reflexive graphs. Journal of Combinatorial Theory (series B), 72(2):236–250, 1998.CrossRefMathSciNetGoogle Scholar
  5. [FHH99]
    Tomas Feder, Pavol Hell, and Jing Huang. List homomorphisms and circular arc graphs. Combinatorica, 19(4):487–505, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [GHN00]
    Anna Galluccio, Pavol Hell, and Jaroslav Nešetřil. The complexity of H- colouring of bounded degree graphs. Discrete Mathematics, 222(1-3):101–109, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [HN90]
    P. Hell and J. Nešetřil. On the complexity of H-coloring. Journal of Combinatorial Theory (series B), 48:92–110, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [TP97]
    Jan Arne Telle and Andrzej Proskurowski. Algorithms for vertex partitioning problems on partial k-trees. SIAM Journal on Discrete Mathematics, 10(4):529–550, 1997.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Josep Díaz
    • 1
  • Maria Serna
    • 1
  • Dimitrios M. Thilikos
    • 1
  1. 1.Departament de Llenguatges i Sistemes InformàticsUniversitat Politècnica de CatalunyaBarcelonaSpain

Personalised recommendations