Unique factorisation of additive induced-hereditary properties

An additive hereditary graph property is a set of graphs, closed under isomorphism and under taking subgraphs and disjoint unions. Let ₁,...,ₙ be additive hereditary graph properties. A graph G has property (₁∘...∘ₙ) if there is a partition (V₁,...,Vₙ) of V(G) into n sets such that, for all i, the induced subgraph ${\displaystyle G\left[V_i\right]}$ is in ${\displaystyle \_i}$. A property is reducible if there are properties , such that = ∘ ; otherwise it is irreducible. Mihók, Semanišin and Vasky [8] gave a factorisation for any additive hereditary property into a given number dc() of irreducible additive hereditary factors. Mihók [7] gave a similar factorisation for properties that are additive and induced-hereditary (closed under taking induced-subgraphs and disjoint unions). Their results left open the possiblity of different factorisations, maybe even with a different number of factors; we prove here that the given factorisations are, in fact, unique.