Covering Graphs by Colored Stable Sets

Let G = ( V , R ∪ B ) be a multigraph with red and blue edges. G is an R / B -split graph if V is the union of a red and a blue stable set. R / B -split graphs yield a common generalization of split graphs and König graphs. It is shown, for example, that R / B -split graphs can be recognized in polynomial time. On the other hand, finding a maximal R / B -subgraph is NP -hard already for the class of comparability graphs of series-parallel orders. Moreover, there can be no approximation ratio better than 31/32 unless P = NP .