The black-and-white coloring problem on distance hereditary graphs and   strongly chordal 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 and |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, distance-hereditary graphs, interval graphs and strongly chordal graphs. We show that the problem is NP-complete on splitgraphs. 
Ton Kloks, Sheung-Hung Poon, Feng-Ren Tsai and Yue-Li Wang
math.CO cs.DS