On the completeness of decomposable properties of graphs

Let ₁,₂ be additive hereditary properties of graphs. A (₁,₂)-decomposition of a graph G is a partition of E(G) into sets E₁, E₂ such that induced subgraph $G[E_i]$ has the property $_i$, i = 1,2. Let us define a property ₁⊕₂ by {G: G has a (₁,₂)-decomposition}. A property D is said to be decomposable if there exists nontrivial additive hereditary properties ₁, ₂ such that D = ₁⊕₂. In this paper we determine the completeness of some decomposable properties and we characterize the decomposable properties of completeness 2