We consider the graph sandwich problem and introduce almost monotone properties, for which the sandwich problem can be reduced to the recognition problem. We show that the property of containing a graph in as an induced subgraph is almost monotone if is the set of thetas, the set of pyramids, or the set of prisms and thetas. We show that the property of containing a hole of length is almost monotone if and only if or . Moreover, we show that the imperfect graph sandwich problem, also known as the Berge trigraph recognition problem, can be solved in polynomial time. We also study the complexity of several graph decompositions related to perfect graphs, namely clique cutset, (full) star cutset, homogeneous set, homogeneous pair, and 1-join, with respect to the partitioned and unpartitioned probe problems. We show that the clique cutset and full star cutset unpartitioned probe problems are NP-hard. We show that for these five decompositions, the partitioned probe problem is in P, and the homogeneous set, 1-join, 1-join in the complement, and full star cutset in the complement unpartitioned probe problems can be solved in polynomial time as well.