  For the edge ideal I of an arbitrary simple graph G we describe the monomials
of the saturation of a power of I in terms of (vertex) weighted graphs
associated with the monomials. This description allows us to characterize the
embedded associated primes of a power of I as covers of G which contain certain
types of subgraphs of G. As an application, we completely classify the
associated primes of the second and the third power of I in terms of G.
