Critical and infinite directed graphs

Given a directed graph G=(V(G), A(G)), a subsetXof V(G) is an interval of Gprovided that for any a, b is an element of X and x is an element of V(G) - X, (a, x) is an element of A(G) if and only if (b, x) is an element of A(G), and similarly for (x, a) and (x, b). For example, phi, {x} (x is an element of V(G)) and V (G) are intervals of G, called trivial intervals. A directed graph is indecomposable if all its intervals are trivial; otherwise, it is decomposable. An indecomposable directed graph G is then critical if for each x is an element of V(G), G (V (G) - {x}) is decomposable and if there are x not equal y is an element of V(G) such that G (V(G) - {x, y}) is indecomposable. A generalization of the lexicographic sum is introduced to describe a process of construction of the critical and infinite directed graphs. It follows that for every critical and infinite directed graph G, there are x not equal y is an element of V(G) such that G and G(V(G) - {x, y}) are isomorphic. It is then deduced that if G is an indecomposable and infinite directed graph and if there is a finite subset F of V(G) such that vertical bar F vertical bar >= 2 and G(V(G) - F) is indecomposable, then there are x not equal y is an element of V(G) such that G(V(G) - {x, y}) is indecomposable. (c) 2007 Elsevier B.V. All rights reserved.