Minimal indecomposable graphs Original Research Article

Let G = (V, E) be a graph, a subset X of V is an interval of G whenever for a, b ∈ X and x ∈ V − X, (a, x) ∈ E (resp. (x, a) ∈ E) if and only if (b, x) ∈ E (resp. (x, b) ∈ E). For instance, 0, x, where x ∈ V, and V are intervals of G, called trivial intervals. A graph G is then said to be indecomposable when all of its intervals are trivial. In the opposite case, we will say that G is decomposable. We now introduce the minimal indecomposable graphs in the following way. Given an indecomposable graph G = (V, E) and vertices x1, …, xk of G, G is said to be minimal for x1, …, xk whenever for every proper subset W of V, if x1, …, xk ∈ W and if |W| ⩾ 3, then the induced subgraph G(W) of G is decomposable. In this paper, we characterize the minimal indecomposable graphs for one or two vertices and we describe in a more precise manner the minimal indecomposable symmetric graphs, posets and tournaments.