Weakly P-saturated graphs

For a hereditary property P let kP(G) denote the number of forbidden subgraphs contained in G. A graph G is said to be weakly Psaturated, if G has the property P and there is a sequence of edges of G, say e1, e2, . . . , el, such that the chain of graphs G = G0 ⊂ G0+e1 ⊂ G1 + e2 ⊂ . . . ⊂ Gl−1 + el = Gl = Kn (Gi+1 = Gi + ei+1) has the following property: kP(Gi+1) > kP(Gi), 0 ≤ i ≤ l − 1. In this paper we shall investigate some properties of weakly saturated graphs. We will find upper bound for the minimum number of edges of weakly Dk-saturated graphs of order n. We shall determine the number wsat(n,P) for some hereditary properties.