For a fixed graph F, a graph G is F-saturated if there is no copy of F in G, but for any edge e ∉ G, there is a copy of F in G + e. The minimum number of edges in an F-saturated graph of order n will be denoted by sat(n, F). A graph G is weakly F-saturated if there is an ordering of the missing edges of G so that if they are added one at a time, each edge added creates a new copy of F. The minimum size of a weakly F-saturated graph G of order n will be denoted by wsat(n, F). The graphs of order...

For a hereditary property let $k_{}\left(G\right)$ denote the number of forbidden subgraphs contained in G. A graph G is said to be weakly -saturated, if G has the property and there is a sequence of edges of G̅, say $e₁,e₂,...,e_l$, such that the chain of graphs $G=G₀\subset G_0+e₁\subset G₁+e₂\subset ...\subset G_{l-1}+e_l=G_l=K_n(G_{i+1}=G_i+e_{i+1})$ has the following property: $k_{}\left(G_{i+1}\right)>k_{}\left(G_i\right)$, 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 ₖ-saturated graphs of order n. We shall determine the number wsat(n,) for some...

The Wiener index of a connected graph is defined as the sum of the distances between all unordered pairs of its vertices. We characterize the graphs which extremize the Wiener index among all graphs on $n$ vertices with $k$ pendant vertices. We also characterize the graph which minimizes the Wiener index over the graphs on $n$ vertices with $s$ cut-vertices.