For any given graphs G and H, we write to mean that any red-blue coloring of the edges of F implies that either F contains a red subgraph G or a blue subgraph H. Graph F is called a Ramsey -minimal graph if but for any proper subgraph The class of all -minimal graphs is denoted by In this paper, we investigate the members of for We prove that is the only disconnected graph in this set and there is no tree as a member. We also prove the uniqueness of such Ramsey minimal graphs with certain girths. In particular, for and 5 we characterize all graphs in