Hierarchical Complexity of 2-Clique-Colouring Weakly Chordal Graphs and Perfect Graphs Having Cliques of Size at Least 3
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چکیده
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-cliquecolouring of perfect graphs is a Σ2 -complete problem [J. Graph Theory 62 (2009) 139–156]. We strengthen this result by showing that it is still Σ2 -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 Σ2 -complete, NP-complete, and P. We solve an open problem posed by Kratochv́ıl 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 Σ2 -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 Σ2 -complete, NP-complete, and P.
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تاریخ انتشار 2014