On the problem of finding small subdivision and homomorphism bases for classes of countable graphs

Let G be a class of countable graphs given by a set Г of forbidden configurations. We consider the following problem: for which Γ is G well characterized by the simplicial decompositions of its members into prime graphs, that is for which G is it possible to find a small subset ℬ of G such that all graphs of G can be constructed from elements of ℬ by successive amalgamations identifying complete subgraphs?