On Cocolourings and Cochromatic Numbers of Graphs

A cocolouring of a graph G is a partition of the vertices such that each set of the partition induces either a clique or an independent set in G. The cochromatic number of G is the smallest cardinality of a cocolouring of G. We show that the cochromatic number problem remains NP-complete for line graphs of comparability graphs (and hence for line graphs and K,, ,-free graphs), and we present polynomial time algorithms for computing the cochromatic numbers of chordal graphs and cographs.