The game of FF-saturator

Let GG be a graph and FF be a family of graphs. We say that GG is FF-saturated if GG does not contain a copy of any member of FF, but for any pair of nonadjacent vertices xx and yy in GG, G+xyG+xy contains a copy of some H∈FH∈F. A great deal of study has been devoted to the maximum and the minimum number of edges in an FF-saturated graph. Little is known, however, about FF-saturated graphs of other sizes or, alternatively, about the FF-saturated subgraphs of an arbitrary graph GG.In this paper, we introduce the game of FF-saturator on GG as a means to study the characteristics of all of the FF-saturated subgraphs of GG. Two players alternate, selecting edges from GG and considering the subgraph of GG formed by all of the edges selected thusfar. The first player to create a copy of HH, for some H∈FH∈F, loses the game. Alternatively, the first player to create an FF-saturated subgraph of GG wins the game. We consider the game of FF-saturator for various choices of FF and GG.