Before revision:  We complement the latter result by showing that, for any fixed p\geq 3 the problem becomes NP-complete when restricted to split (S_3,\ldots,S_p)-free graphs, where S_q denotes the q-sun graph.
After revision:  We complement the latter result by showing that, for any fixed p\geq 3 both variants of the problem become NP-complete when restricted to split (S_3,\ldots,S_p)-free graphs, where S_q denotes the q-sun graph.