C4-saturated bipartite graphs

Let H be a graph. A graph G is said to be H-free if it contains no subgraph isomorphic to H. A graph G is said to be an H-saturated subgraph of a graph K if G is an H-free subgraph of K with the property that for any edge e∈E(K)⧹E(G), G∪{e} is not H-free. We present some general results on Ks,t-saturated subgraphs of the complete bipartite graph Km,n and study the problem of finding, for all possible values of q, a C4-saturated subgraph of Km,n having precisely q edges.