Size of weakly saturated graphs

Let PP be a hereditary property. Let kP(G)kP(G) denote the number of forbidden subgraphs, which are contained in G. A graph G is said to be weakly  PP-saturated  , if G∈PG∈P and the edges of the complement of G   can be labelled e1,e2,…,ele1,e2,…,el in such way that for i=0,1,…,l-1i=0,1,…,l-1 the inequality kP(Gi+1)>kP(Gi)kP(Gi+1)>kP(Gi) holds, where G0=G,Gi+1=Gi+eiG0=G,Gi+1=Gi+ei and Gl=KnGl=Kn. The minimum possible size of weakly PP-saturated graphs is denoted by wsat(n,P)wsat(n,P).In this paper we shall investigate some properties of weakly saturated graphs. We provide some estimations for the minimum size of weakly PP-saturated graphs.