Partitioning graphs into complete and empty graphs

Given integers jj and kk and a graph GG, we consider partitions of the vertex set of GG into j+kj+k parts where jj of these parts induce empty graphs and the remaining kk induce cliques. If such a partition exists, we say GG is a (j,k)(j,k)-graph. For a fixed jj and kk we consider the maximum order nn where every graph of order nn is a (j,k)(j,k)-graph. The split-chromatic number of GG is the minimum jj where GG is a (j,j)(j,j)-graph. Further, the cochromatic number is the minimum j+kj+k where GG is a (j,k)(j,k)-graph. We examine some relations between cochromatic, split-chromatic and chromatic numbers. We also consider some computational questions related to chordal graphs and cographs.