Hierarchical complexity of 2-clique-colouring weakly chordal graphs and perfect graphs having cliques of size at least 3

الأصل · EN

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 Σ₂ᵖ-complete problem [J. Graph Theory 62 (2009) 139--156]. We strengthen this result by showing that it is still Σ₂ᵖ-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 Σ₂ᵖ-complete, NP-complete, and 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 Σ₂ᵖ-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 Σ₂ᵖ-complete, NP-complete, and P.