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. D´efossez proved that the 2-cliquecolouring of perfect graphs is a ΣP 2 -complete problem [J. Graph Theory 62 (2009) 139–156]. We strengthen this result by showing that it is still ΣP 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 ΣP 2 -complete, NP-complete, and P. We solve an open problem posed by Kratochv´ıl 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 ΣP 2 -complete problem. We then determine a hierarchy of nested subclasses of perfect graphs with all cliques having size at least 3 whereby each...

• Type: Journal
• Year: 2016
• Where: TCS
• Authors: Hélio B. Macêdo Filho, Raphael C. S. Machado, Celina M. Herrera de Figueiredo
• Keywords: TCS 2016, Theoretical Computer Science