The complexity of forbidden subgraph sandwich problems and the skew partition sandwich problem

The ? graph sandwich problem asks, for a pair of graphs G 1 = ( V , E 1 ) and G 2 = ( V , E 2 ) with E 1 ? E 2 , whether there exists a graph G = ( V , E ) that satisfies property ? and E 1 ? E ? E 2 . We consider the property of being F -free, where F is a fixed graph. We show that the claw-free graph sandwich and the bull-free graph sandwich problems are both NP-complete, but the paw-free graph sandwich problem is polynomial. This completes the study of all cases where F has at most four vertices. A skew partition of a graph G is a partition of its vertex set into four nonempty parts A , B , C , D such that each vertex of A is adjacent to each vertex of B , and each vertex of C is nonadjacent to each vertex of D . We prove that the skew partition sandwich problem is NP-complete, establishing a computational complexity non-monotonicity.