Given a directed graph G=(V,A), the induced subgraph of G by a subset X of V is denoted by G[X]. A subset X of V is an interval of G provided that for a,b∈X and x∈V∖X, (a,x)∈A if and only if (b,x)∈A, and similarly for (x,a) and (x,b). For instance, 0̸, V and {x}, x∈V, are intervals of G, called trivial intervals. A directed graph is indecomposable if all its intervals are trivial, otherwise it is decomposable. Given an indecomposable directed graph G=(V,A), a vertex x of G is critical if G[V∖{x}] is decomposable. An indecomposable directed graph is critical when all its vertices are critical. With each indecomposable directed graph G=(V,A) is associated its indecomposability directed graph Ind(G) defined on V by: given x≠y∈V, (x,y) is an arc of Ind(G) if G[V∖{x,y}] is indecomposable. All the results follow from the study of the connected components of the indecomposability directed graph. First, we prove: if G is an indecomposable directed graph, which admits at least two non critical vertices, then there is x∈V such that G[V∖{x}] is indecomposable and non critical. Second, we characterize the indecomposable directed graphs G which have a unique non critical vertex x and such that G[V∖{x}] is critical. Third, we propose a new approach to characterize the critical directed graphs.