  Motivated by Chudnovsky's structure theorem of bull-free graphs, Abu-Khzam,
Feghali, and M\"uller have recently proved that deciding if a graph has a
vertex partition into disjoint cliques and a triangle-free graph is NP-complete
for five graph classes. The problem is trivial for the intersection of these
five classes. We prove that the problem is NP-complete for the intersection of
two subsets of size four among the five classes. We also show NP-completeness
for other small classes, such as graphs with maximum degree 4 and line graphs.
