Reducible properties of graphs

Let L be the set of all hereditary and additive properties of graphs. For P₁, P₂ ∈ L, the reducible property R = P₁∘P₂ is defined as follows: G ∈ R if and only if there is a partition V(G) = V₁∪ V₂ of the vertex set of G such that ${\displaystyle ⟨V₁{⟩}_G\in P₁}$ and ${\displaystyle ⟨V₂{⟩}_G\in P₂}$. The aim of this paper is to investigate the structure of the reducible properties of graphs with emphasis on the uniqueness of the decomposition of a reducible property into irreducible ones.