Indecomposability graph and critical vertices of an indecomposable graph

Abstract

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.