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(P)(2)-complete problem [J. Graph Theory 62 ( 2009) 139-156]. We strengthen this result by showing that it is still Sigma(P)(2)-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(P)(2)-complete, NP-complete, and 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(P)(2)-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(P)(2)-complete, NP-complete, and P.