Triangle-free graphs which are minimal for some nonstable 4-vertex subset

In a graph GG, a module   is a vertex subset MM such that every vertex outside MM is adjacent to all or none of MM. A graph GG is prime   if ϕϕ, the single-vertex sets, and V(G)V(G) are the only modules in GG. A prime graph GG is kk-minimal   if there is some kk-set UU of vertices such that no proper induced subgraph of GG containing UU is prime.Cournier and Ille in 1998 characterized the 1-minimal and 2-minimal graphs. Recently, Alzohairi and Boudabbous characterized 3-minimal triangle-free graphs. We characterize the triangle-free graphs which are minimal for some nonstable 4-vertex subset.