Weak saturation numbers for multiple copies

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 G contains no copy of F and 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 , F ) for many families of vertex disjoint copies of connected graphs such as small order graphs, trees, cycles, and complete graphs will be determined.