NP-hardness results for partitioning graphs into disjoint cliques and a   triangle-free subgraph
  This paper investigates the computational complexity of deciding whether the vertices of a graph can be partitioned into a disjoint union of cliques and a triangle-free subgraph. This problem is known to be $\NP$-complete on arbitrary graphs. We show that this problem remains $\NP$-complete even when restricted to planar graphs and perfect graphs. 
Carl Feghali, Faisal N. Abu-Khzam, Haiko M\"uller
cs.DM cs.CC