NP-hardness of the recognition of coordinated graphs

  • Francisco J. Soulignac
  • Gabriel Sueiro


A graph G is coordinated if the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of H with a common vertex, for every induced subgraph H of G. In previous works, polynomial time algorithms were found for recognizing coordinated graphs within some classes of graphs. In this paper we prove that the recognition problem for coordinated graphs is NP-hard, and it is NP-complete even when restricted to the class of {gem, C 4, odd hole}-free graphs with maximum degree four, maximum clique size three and at most three cliques sharing a common vertex.


Computational complexity Coordinated graph recognition {gem, C4, odd hole}-free graphs NP-complete problems 


  1. Bonomo, F. (2005). On subclasses and variations of perfect graphs. PhD thesis, Departmento de Computación, FCEyN, Universidad de Buenos Aires, Buenos Aires. Google Scholar
  2. Bonomo, F., Durán, G., Groshaus, M., & Szwarcfiter, J. L. (2006a). On clique-perfect and K-perfect graphs. Ars Combinatoria, 80, 97–112. Google Scholar
  3. Bonomo, F., Durán, G., Soulignac, F., & Sueiro, G. (2006b). Partial characterizations of coordinated graphs: line graphs and complements of forests. In ISMP’06. Google Scholar
  4. Bonomo, F., Durán, G., & Groshaus, M. (2007). Coordinated graphs and clique graphs of clique-Helly perfect graphs. Utilitas Mathematica, 72, 175–191. Google Scholar
  5. Bonomo, F., Chudnovsky, M., & Durán, G. (2008a). Partial characterizations of clique-perfect graphs, I: sublcasses of claw-free graphs. Discrete Applied Mathematics, 156(7), 1058–1082. CrossRefGoogle Scholar
  6. Bonomo, F., Durán, G., Soulignac, F., & Sueiro, G. (2008b). Partial characterizations of clique-perfect and coordinated graphs: superclasses of triangle-free graphs. Electronic Notes in Discrete Mathematics, 30, 51–56. CrossRefGoogle Scholar
  7. Chudnovsky, M., Cornuéjols, G., Liu, X., Seymour, P., & Vušković, K. (2005). Recognizing Berge graphs. Combinatorica, 25(2), 143–186. CrossRefGoogle Scholar
  8. Chudnovsky, M., Robertson, N., Seymour, P., & Thomas, R. (2006). The strong perfect graph theorem. Annals of Mathematics (2), 164(1), 51–229. CrossRefGoogle Scholar
  9. Escalante, F. (1973). Über iterierte Clique-Graphen. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 39, 59–68. Google Scholar
  10. Golumbic, M. C. (2004). Annals of discrete mathematics: Vol. 57. Algorithmic graph theory and perfect graphs (2nd. ed.). Amsterdam: North-Holland. Google Scholar
  11. Grötschel, M., Lovász, L., & Schrijver, A. (1981). The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1(2), 169–197. CrossRefGoogle Scholar
  12. Guruswami, V., & Rangan, C. P. (2000). Algorithmic aspects of clique-transversal and clique-independent sets. Discrete Applied Mathematics, 100(3), 183–202. CrossRefGoogle Scholar
  13. Maffray, F., & Preissmann, M. (1996). On the NP-completeness of the k-colorability problem for triangle-free graphs. Discrete Mathematics, 162(1–3), 313–317. CrossRefGoogle Scholar
  14. Soulignac, F., & Sueiro, G. (2006). Exponential families of minimally non-coordinated graphs. In The 2nd Latin-American workshop on cliques in graphs. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Departamento de Computación, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina

Personalised recommendations