Graphs having the local decomposition property

Let H be a fixed graph without isolated vertices, and let G be a graph on n vertices. Let 2 ≤ k ≤ n - 1 be an integer. We prove that if k ≤ n - 2 and every k-vertex induced subgraph of G is H-decomposable then G or its complement is either a complete graph or a complete bipartite graph. This also holds for k = n - 1 if all the degrees of the vertices of H have a common factor. On the other hand, we show that there are graphs H for which it is NP-Complete to decide if every n - 1-vertex subgraph of G is H-decomposable. In particular, we show that H = K_1,h-1, where h > 3, are such graphs.