Geodesic
Unique factorisation of additive induced-hereditary properties
Discussiones Mathematicae. Graph Theory, Tome 24 (2004) no. 2, pp. 319-343

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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 G[V_i] is in _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.
Keywords: additive and hereditary graph classes, unique factorization 
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Farrugia, Alastair; Richter, R. Unique factorisation of additive induced-hereditary properties. Discussiones Mathematicae. Graph Theory, Tome 24 (2004) no. 2, pp. 319-343. http://geodesic.mathdoc.fr/item/DMGT_2004_24_2_a12/