Decomposition tree and indecomposable coverings

Let G = (V,A) be a directed graph. With any subset X of V is associated the directed subgraph G[X ] = (X,A ∩ (X × X)) of G induced by X . A subset X of V is an interval of G provided that for a, b ∈ X and x ∈ V \ X , (a, x) ∈ A if and only if (b, x) ∈ A, This work was supported, in part, by an NSF EPSCoR grant EPS-0346476 and by a Nebraska Research Initiative (NRI) grant on high performance wireless networks. This research was done while the third author was visiting the University of Nebraska – Lincoln. 38 A. Breiner, J. Deogun and P. Ille and similarly for (x, a) and (x, b). For example ∅, V, and {x}, where x ∈ V , are intervals of G which are the trivial intervals. A directed graph is indecomposable if all its intervals are trivial. Given an integer k > 0, a directed graph G = (V,A) is called an indecomposable kcovering provided that for every subset X of V with |X | ≤ k, there exists a subset Y of V such that X ⊆ Y , G[Y ] is indecomposable with |Y | ≥ 3. In this paper, the indecomposable k-covering directed graphs are characterized for any k > 0.