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 ${\cal P}_1, >..., {\cal P}_n$ be additive hereditary graph properties. A graph $G$ has property $({\cal P}_1 \circ ... \circ {\cal P}_n)$ if there is a partition $(V_1, ..., V_n)$ of $V(G)$ into $n$ sets such that, for all $i$, the induced subgraph $G[V_i]$ is in ${\cal P}_i$. A property ${\cal P}$ is reducible if there are properties ${\cal Q}$, ${\cal R}$ such that ${\cal P} = {\cal Q} \circ {\cal R}$; otherwise it is irreducible. Mih\'{o}k, Semani\v{s}in and Vasky [J. Graph Theory {\bf 33} (2000), 44--53] gave a factorisation for any additive hereditary property ${\cal P}$ into a given number $dc({\cal P})$ of irreducible additive hereditary factors. Mih\'{o}k [Discuss. Math. Graph Theory {\bf 20} (2000), 143--153] 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.