Indecomposability graph and critical vertices of an indecomposable graph

Given a directed graph G=(V,A)G=(V,A), the induced subgraph of GG by a subset XX of VV is denoted by G[X]G[X]. A subset XX of VV is an interval of GG provided that for a,b∈Xa,b∈X and x∈V∖Xx∈V∖X, (a,x)∈A(a,x)∈A if and only if (b,x)∈A(b,x)∈A, and similarly for (x,a)(x,a) and (x,b)(x,b). For instance, 0̸0̸, VV and {x}{x}, x∈Vx∈V, are intervals of GG, 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)G=(V,A), a vertex xx of GG is critical if G[V∖{x}]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)G=(V,A) is associated its indecomposability directed graph Ind(G) defined on VV by: given x≠y∈Vx≠y∈V, (x,y)(x,y) is an arc of Ind(G) if G[V∖{x,y}]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 GG is an indecomposable directed graph, which admits at least two non critical vertices, then there is x∈Vx∈V such that G[V∖{x}]G[V∖{x}] is indecomposable and non critical. Second, we characterize the indecomposable directed graphs GG which have a unique non critical vertex xx and such that G[V∖{x}]G[V∖{x}] is critical. Third, we propose a new approach to characterize the critical directed graphs.