Number of ways to bicolor a graph with some constriants
If you only care about graphs with at most $32$ vertices then just list all possible colorings. They are only a constant number ($2^{32}$). For each of them you can decide in constant time whether it is valid according to your constraints.
If you care about graphs with any number of vertices, then the problem is a generalization of the problem of counting the number of dominating sets in a graph, which is #P-complete even for special classes of graphs.