Abstract
In a graph G, a module is a vertex subset M such that every vertex outside M is adjacent to all or none of M. A graph G is prime if ϕ, the single-vertex sets, and V(G) are the only modules in G. A prime graph G is k-minimal if there is some k-set U of vertices such that no proper induced subgraph of G containing U 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.
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