**Description of the Triangle-free Prime Graphs Having at Most Two Non-Critical Vertices**

Imed Boudabbous and Walid Marweni

In a graph *G* = (*V*, *E*), a module is a vertex subset *M* such that every vertex outside *M* is adjacent to all or none of *M*. For example, ∅, {*x*} (*x* ∈ *V*) and *V* are modules of *G*, called trivial modules. A graph, all the modules of which are trivial, is prime; otherwise, it is decomposable. A vertex *x* of a prime graph *G* is critical if *G* − *x* is decomposable. We generalize this definition. A prime graph *G* = (*V*, *E*) is *X*-critical, where *X* is a subset of *V*, if for each *x* ∈ *X*, *x* is a critical vertex. Moreover, *G* is (−*k*)-critical, where 0 ≤ *k* ≤ |*V*|, if *G* is (*V*\*X*)-critical for some subset *X* of *V* such that |*X*| = *k*. In 1993, J.H. Schmerl and W.T. Trotter characterized the *V*-critical graphs, called critical graphs with vertex set *V*. Recently, H. Belkhechine, I. Boudabbous and M.B. Elayech characterized the (−1)-critical graphs. In this paper, we characterize the (−*k*)-critical graphs within the family of triangle-free graphs where *k* ≤ 2.

*Keywords:* Graphs, prime, module, critical vertex, partially critical