Hierarchical Complexity of 2-Clique-Colouring Weakly Chordal Graphs and Perfect Graphs Having Cliques of Size at Least 3

  • Helio B. Macêdo Filho
  • Raphael C. S. Machado
  • Celina M. H. Figueiredo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)


A clique of a graph is a maximal set of vertices of size at least 2 that induces a complete graph. A k-clique-colouring of a graph is a colouring of the vertices with at most k colours such that no clique is monochromatic. Défossez proved that the 2-clique-colouring of perfect graphs is a \(\Sigma_2^P\)-complete problem [J. Graph Theory 62 (2009) 139–156]. We strengthen this result by showing that it is still \(\Sigma_2^P\)-complete for weakly chordal graphs. We then determine a hierarchy of nested subclasses of weakly chordal graphs whereby each graph class is in a distinct complexity class, namely \(\Sigma_2^P\)-complete, \(\mathcal{NP}\)-complete, and \(\mathcal{P}\). We solve an open problem posed by Kratochvíl and Tuza to determine the complexity of 2-clique-colouring of perfect graphs with all cliques having size at least 3 [J. Algorithms 45 (2002), 40–54], proving that it is a \(\Sigma_2^P\)-complete problem. We then determine a hierarchy of nested subclasses of perfect graphs with all cliques having size at least 3 whereby each graph class is in a distinct complexity class, namely \(\Sigma_2^P\)-complete, \(\mathcal{NP}\)-complete, and \(\mathcal{P}\).


(α, β)-polar graphs clique-colouring hierarchical complexity perfect graphs weakly chordal graphs 


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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Helio B. Macêdo Filho
    • 1
  • Raphael C. S. Machado
    • 2
  • Celina M. H. Figueiredo
    • 1
  1. 1.COPPEUniversidade Federal do Rio de JaneiroBrazil
  2. 2.Inmetro — Instituto Nacional de Metrologia, Qualidade e TecnologiaBrazil

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