List Partitions of Chordal Graphs

  • Tomás Feder
  • Pavol Hell
  • Sulamita Klein
  • Loana Tito Nogueira
  • Fábio Protti
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2976)


In an earlier paper we gave efficient algorithms for partitioning chordal graphs into k independent sets and ell cliques. This is a natural generalization of the problem of recognizing split graphs, and is NP-complete for graphs in general, unless k ≤ 2 and ell ≤ 2. (Split graphs have k = ell = 1.)

In this paper we expand our focus and consider general M-partitions, also known as trigraph homomorphisms, for the class of chordal graphs. For each symmetric matrix M over 0, 1, *, the M-partition problem seeks a partition of the input graph into independent sets, cliques, or arbitrary sets, with some pairs of sets being required to have no edges, or to have all edges joining them, as encoded in the matrix M. Such partitions generalize graph colorings and homomorphisms, and arise frequently in the theory of graph perfection. We show that many M-partition problems that are NP-complete in general become solvable in polynomial time for chordal graphs, even in the presence of lists. On the other hand, we show that there are M-partition problems that remain NP-complete even for chordal graphs. We also discuss forbidden subgraph characterizations for the existence of M-partitions.


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  1. 1.
    Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21, 358–402 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cameron, K., Eschen, E.M., Hoang, C.T., Sritharan, R.: The list partition problem for graphs. In: SODA 2004 (2004)Google Scholar
  3. 3.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem (manuscript 2002)Google Scholar
  4. 4.
    de Figueiredo, C.M.H., Klein, S., Kohayakawa, Y., Reed, B.A.: Finding skew partitions efficiently. Journal of Algorithms 37, 505–521 (2000)zbMATHCrossRefGoogle Scholar
  5. 5.
    Diaz, J., Serna, M., Thilikos, D.M.: The complexity of parametrized Hcolorings: a survey. Dimacs Series in Discrete Mathematics (2003)Google Scholar
  6. 6.
    Feder, T., Hell, P.: List constraint satisfaction and list partition. Submitted to SIAM J. Comput. (2003)Google Scholar
  7. 7.
    Feder, T., Hell, P., Huang, J.: List homomorphisms and circular arc graphs. Combinatorica 19, 487–505 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Feder, T., Hell, P., Klein, S., Motwani, R.: Complexity of list partitions. In: Proc. 31st Ann. ACM Symp. on Theory of Computing, pp. 464–472 (1999); SIAM J. Comput. (in press)Google Scholar
  9. 9.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)zbMATHGoogle Scholar
  10. 10.
    Hell, P., Klein, S., Nogueira, L.T., Protti, F.: Partitioning chordal graphs into independent sets and cliques. Discrete Applied Math. (to appear)Google Scholar
  11. 11.
    Proskurowski, A., Arnborg, S.: Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Applied Math. (23), 11–24 (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Tomás Feder
    • 1
  • Pavol Hell
    • 2
  • Sulamita Klein
    • 3
  • Loana Tito Nogueira
    • 4
  • Fábio Protti
    • 5
  1. 1. Palo AltoUSA
  2. 2.School of Computing ScienceSimon Fraser UniversityBurnabyCanada
  3. 3.IM and COPPE-SistemasUniversidade Federal do Rio de Janeiro CaixaRio de JaneiroBrasil
  4. 4.COPPE-SistemasUniversidade Federal do Rio de Janeiro CaixaRio de JaneiroBrasil
  5. 5.IM and NCEUniversidade Federal do Rio de Janeiro CaixaRio de JaneiroBrasil

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