3-minimal triangle-free graphs

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 11-minimal and 22-minimal graphs. We characterize 33-minimal triangle-free graphs. As a corollary, we show that there are exactly [(n−1)212]−⌊n−42⌋+⌊n−22⌋ nonisomorphic 33-minimal triangle-free nn-vertex graphs when n≥7n≥7, where [x][x] denotes the nearest integer to xx.