max-cut and containment relations in graphs

We study max-cut in classes of graphs defined by forbidding finitely many graphs as subgraphs, or a single graph as an induced subgraph or a minor. For the first two containment relations, we prove dichotomy theorems. For the minor order, we show how to solve max-cut in polynomial time for the class obtained by forbidding a graph with a single crossing (this generalizes a known result for K5-minor-free graphs) and identify an open problem which is the missing case for a dichotomy theorem.