Indecomposable graphs

Abstract

Let G = ( V, E) be a finite directed graph. A subset X of V is an interval of G if for a, b ∈ X and x ∈ V − X, we have ax ∈ E (resp. xa ∈ E) if and only if bx ∈ E (resp. xb ∈ E). So ∅, V and every singleton are intervals of G (called trivial intervals). The graph G is said to be indecomposable if every interval is trivial. In this work, we study the induced subgraphs of an indecomposable graph which are also indecomposable. In particular, we prove: Theorem. Let G = ( V, E) be an indecomposable graph and let X be a subset of V such that | X| ⩾ 3, | V − X| ⩾ 6 and G( X) is indecomposable. Then there is a subset Y of V fulfilling X ⊆ Y, | V − Y| = 2 and G( Y) is indecomposable.