To our best knowledge, all existing graph pattern mining algorithms can only mine either closed, maximal or the complete set of frequent subgraphs instead of graph generators which are preferable to the closed subgraphs according to the Minimum Description Length principle in some applications. In this paper, we study a new problem of frequent subgraph mining, called frequent connected graph generator mining, which poses significant challenges due to the underlying complexity associated with frequent subgraph mining as well as the absence of Apriori property for graph generators. Whereas, we still present an efficient solution Fogger1 for this new problem. By exploring some properties of graph generators, two effective pruning techniques, backward edge pruning and forward edge pruning, are proposed to prune the branches of the well-known DFS code enumeration tree that do not contain graph generators. To further improve the efficiency, an effective index structure, ADI++, is also de...