Minimal indecomposable graphs

Let G=(V,E) be a graph, a subset X of V is an interval of G whenever for a, b E X and xE V X , (a,x)EE (resp. (x,a)EE) if and only if (b,x)EE (resp. (x,b)EE). For instance, 0, {x}, where x E 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 xl , . . . ,xk of G, G is said to be minimal for xl,.. . ,xk whenever for every proper subset W of V, if Xl,...,xk E 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.