INFINITE INDECOMPOSABLE GRAPHS

Let G = (V, E) be a directed graph. A subset X of V is an interval of G if for all the elements a, b of X and x of V-X, we have: (a, x) is-a n-element-of E [resp. (x, a) is-an-element-of E] if and only if (b, x) is-an-element-of E [resp. (x, b) is-an-element-of E]. So phi, V and e very singleton are intervals of G (called trivial intervals). The grap h G is said to be indecomposable if every interval is trivial. In this Note, we examine the induced subgraphs of an infinite indecomposable graph which are also indecomposable. In particular, we prove the follo wing theorem: Let G = (V, E), be an infinite graph, G is indecomposabl e if and only if for each finite subset X of V, there is a finite subs et Y of V containing X and such that the induced subgraph of G(Y) G is indecomposable.