Clique Graph Recognition Is NP-Complete

  • L. Alcón
  • L. Faria
  • C. M. H. de Figueiredo
  • M. Gutierrez
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4271)


A complete set of a graph G is a subset of V inducing a complete subgraph. A clique is a maximal complete set. Denote by \({\mathcal{C}}(G)\) the clique family of G. The clique graph of G, denoted by K(G), is the intersection graph of \(\mathcal{{C}}(G)\). Say that G is a clique graph if there exists a graph H such that G=K(H). The clique graph recognition problem asks whether a given graph is a clique graph. A sufficient condition was given by Hamelink in 1968, and a characterization was proposed by Roberts and Spencer in 1971. We prove that the clique graph recognition problem is NP-complete.


Planar Graph Complete Graph Intersection Graph Truth Assignment Nonempty Intersection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alcón, L., Gutierrez, M.: A new characterization of Clique Graphs. Matemática Contemporânea 25, 1–7 (2003)MATHGoogle Scholar
  2. 2.
    Alcón, L., Gutierrez, M.: Cliques and Extended Triangles. A necessary condition to be Clique Planar Graph. Discrete Applied Mathematics 141/1-3, 3–17 (2004)CrossRefGoogle Scholar
  3. 3.
    Berge, C.: Hypergraphes. Gauthier-Villars Paris (1987)Google Scholar
  4. 4.
    Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A survey. SIAM Monographs on Discrete Mathematics and Applications (1999)Google Scholar
  5. 5.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: a guide to the theory of NP-completeness. W.H. Freeman, New York (1979)MATHGoogle Scholar
  6. 6.
    Hamelink, R.C.: A partial characterization of clique graphs. Journal of Combinatorial Theory B 5, 192–197 (1968)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Papadimitriou, C.M.: Computational Complexity. Addison-Wesley, Reading (1994)MATHGoogle Scholar
  8. 8.
    Prisner, E.: A common generalization of Line Graphs and Clique Graphs. Journal of Graph Theory 18, 301–313 (1994)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Roberts, F.S., Spencer, J.H.: A characterization of clique graphs. Journal of Combinatorial Theory B 10, 102–108 (1971)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Szwarcfiter, J.L.: A survey on Clique Graphs. In: Linhares-Sales, C., Reed, B. (eds.) Recent Advances in Algorithmic Combinatorics. Springer, Heidelberg (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • L. Alcón
    • 1
  • L. Faria
    • 2
  • C. M. H. de Figueiredo
    • 3
  • M. Gutierrez
    • 1
  1. 1.Departamento de MatemáticaUNLPArgentina
  2. 2.Departamento de MatemáticaFFP, UERJBrazil
  3. 3.Instituto de Matemática and COPPEUFRJBrazil

Personalised recommendations