Generalized coloring for tree-like graphs

  • Klaus Jansen
  • Petra Scheffler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 657)


We discuss the Precoloring Extension (PrExt) and the List Coloring (LiCol) problems for trees, partial k-trees and cographs in the decision and the construction versions. Both problems for partial k-trees are solved in linear time, when the number of colors is a constant and by O(¦V¦k+2)-algorithmsin general. For trees, we improve this to linear time. In contrast to that, PrExt and LiCol differ in complexity for cographs. While the first has a linear algorithm, the second is shown NP-complete. We give polynomial algorithms for the corresponding enumeration problems #PrExt and #LiCol on partial k-trees and trees and for #PrExt on cographs.


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  1. [ALS]
    S. Arnborg, J. Lagergren and D. Seese: Problems easy for tree-decomposable graphs. In: Proc. 15th ICALP, LNCS 317, Springer Berlin, 1988, 38–51Google Scholar
  2. [AP]
    S. Arnborg and A. Proskurowski: Linear-time algorithms for NP-hard problems on graphs embedded in k-trees. Discr. Appl. Math. 23 (1989) 11–24Google Scholar
  3. [BLW]
    M. W. Bern, E. L. Lawler and A. L. Wong: Linear-time computations of subgraphs of decomposable graphs. J. Algorithms 8 (1987), 216–235Google Scholar
  4. [BHT]
    M. Biró, M. Hujter and Zs. Tuza: Precoloring Extension. I. Interval Graphs. Discrete Math., to appearGoogle Scholar
  5. [CPS]
    D. G. Corneil, Y. Perl and L. K. Stewart: A linear recognition algorithm for cographs. SIAM J. Comput. 4 (1985) 926–934Google Scholar
  6. [GJ]
    M. R. Garey and D.S. Johnson: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, 1979Google Scholar
  7. [HT]
    M. Hujter and Zs. Tuza: Precoloring Extension. II. Graph Classes Related to Bipartite graphs. J. Graph Theory, to appearGoogle Scholar
  8. [Jan]
    K. Jansen: Ein Zurdnungproblem im Hardware Design. Dissertation, Universität Trier, FB IV — Mathematik und Informatik, Trier 1990Google Scholar
  9. [Ree]
    B. Reed: Finding approximate separators and computing tree-width quickly. In: Proc. STOC'92, 1992Google Scholar
  10. [RS]
    N. Robertson and P. Seymour: Graph Minors. II. Algorithmic aspects of tree-width. J. Algorithms 7 (1986) 309–322Google Scholar
  11. [Sch]
    P. Scheffler: Die Baumweite von Graphen als ein Maß für die Kompliziertheit algorithmischer Probleme. Report RMATH-04/89, K.-Weierstraß-Institut für Mathematik, Berlin 1989Google Scholar
  12. [Viz]
    V. G. Vizing: Critical graphs with given chromatic class (Russian). Diskret. Analiz 5 (1965) 9–17Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Klaus Jansen
    • 1
  • Petra Scheffler
    • 2
  1. 1.FB IV, Universität TrierTrier
  2. 2.FB 3, TU Berlin, MA 6-1Berlin 12

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