On the complexity of the black-and-white coloring problem on some classes of perfect graphs

Given a graph G and integers b and w. The black-and-white coloring problem asks if there exist disjoint sets of vertices B and W   with |B|=b|B|=b and |W|=w|W|=w such that no vertex in B is adjacent to any vertex in W. In this paper we show that the problem is polynomial when restricted to cographs, interval graphs, permutation graphs, distance-hereditary graphs, and strongly chordal graphs. We show that the problem is NP-complete on splitgraphs.