Weak saturation numbers for sparse graphs

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 6∈ 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 n that are weakly F -saturated will be denoted by wSAT(n, F ), and those graphs in wSAT(n, F ) with wsat(n, F ) edges will be denoted by wSAT(n, F ). The precise value of wsat(n, T ) for many families of sparse graphs, and in particular for many trees, will be determined. More specifically, families of trees for which wsat(n, T ) = |T |−2 will be determined. The maximum and minimum values of wsat(n, T ) for the class of all trees will be given. Some properties of wsat(n, T ) and wSAT(n, T ) for trees will be discussed.