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

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. Defossez 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 Kratochvil 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 \(\mathcal{P}\).