Partitioning a graph into disjoint cliques and a triangle-free graph

A graph G=(V,E) is partitionable if there exists a partition {A,B} of V such that A induces a disjoint union of cliques (i.e., G[A] is P3-free) and B induces a triangle-free graph (i.e., G[B] is K3-free). In this paper we investigate the computational complexity of deciding whether a graph is partitionable. The problem is known to be NP-complete on arbitrary graphs. Here it is proved that if a graph G is bull-free, planar, perfect, K4-free or does not contain certain holes then deciding whether G is partitionable is NP-complete. This answers an open question posed by Thomassé, Trotignon and VuÅ¡ković. In contrast a finite list of forbidden induced subgraphs is given for partitionable cographs.