Recognizing a finite group from the generating properties of its subsets

We assume to have information about the generating properties of the subsets of a finite group G. In particular, we consider the two following situations. We know, for every subset X of G, whether X is a generating set of G. We know the graph whose vertices are the subsets of G and in which there is an edge connecting X and Y if and only if $$X\cup Y$$ is a generating set of G. We discuss how this kind of information can be used to discover properties of the group G.