Indecomposability graph and indecomposability recognition

Given a digraph G = ( V , A ) , a subset X of V is an interval of G if for a , b ∈ X and v ∈ V ∖ X , ( a , v ) ∈ A if and only if ( b , v ) ∈ A , and similarly for ( v , a ) and ( v , b ) . For instance, 0̸ , V and { v } , v ∈ V , are intervals of G called trivial. A digraph is indecomposable if all its intervals are trivial. Let G = ( V , A ) be a digraph. Given v ∈ V , v is an indecomposability vertex of G if G [ V ∖ { v } ] is indecomposable. The indecomposability graph I ( G ) of G is defined on V as follows. Given v ≠ w ∈ V , { v , w } is an edge of I ( G ) if G [ V ∖ { v , w } ] is indecomposable. The following is proved for an indecomposable digraph G = ( V , A ) . For every digraph H = ( V , B ) , if G and H have the same indecomposability vertices and if d I ( G ) ( v ) = d I ( H ) ( v ) for each v ∈ V , then H is indecomposable. We also study other types of indecomposability recognition.