Two graph isomorphism polytopes

The convex hull @j"n","n of certain (n!)^2 tensors was considered recently in connection with graph isomorphism. We consider the convex hull @j"n of the n! diagonals among these tensors. We show: 1. The polytope @j"n is a face of @j"n","n. 2. Deciding if a graph G has a subgraph isomorphic to H reduces to optimization over @j"n. 3. Optimization over @j"n reduces to optimization over @j"n","n. In particular, this implies that the subgraph isomorphism problem reduces to optimization over @j"n","n.