A corrected proof shows that every graph with n vertices and 2n-3 edges and no stable cutset is built from triangles and 6-cycles glued along edges or triangles.
Chv´ atal, Recognizing decomposable graphs, Journal of Gra ph Theory 8 (1984) 51–53
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Revisiting Extremal Graphs Having No Stable Cutsets
A corrected proof shows that every graph with n vertices and 2n-3 edges and no stable cutset is built from triangles and 6-cycles glued along edges or triangles.