Criteria for of the existence of uniquely partitionable graphs with respect to additive induced-hereditary properties

Let ₁,₂,...,ₙ be graph properties, a graph G is said to be uniquely (₁,₂, ...,ₙ)-partitionable if there is exactly one (unordered) partition {V₁,V₂,...,Vₙ} of V(G) such that $G[V_i] ∈ _i$ for i = 1,2,...,n. We prove that for additive and induced-hereditary properties uniquely (₁,₂,...,ₙ)-partitionable graphs exist if and only if $_i$ and $_j$ are either coprime or equal irreducible properties of graphs for every i ≠ j, i,j ∈ {1,2,...,n}