The max-cut problem is a fundamental and much-studied NP-hard combinatorial optimisation problem, with a wide range of applications. Several authors have shown that the max-cut problem can be solved in polynomial time if the underlying graph is free of certain minors. We give a polyhedral counterpart of these results. In particular, we show that, if a family of valid inequalities for the cut polytope satisfies certain conditions, then there is an associated minor-closed family of graphs on which the max-cut problem can be solved efficiently.