Unique factorization theorem

A property of graphs is any class of graphs closed under isomorphism. A property of graphs is induced-hereditary and additive if it is closed under taking induced subgraphs and disjoint unions of graphs, respectively. Let P1,P2, . . . ,Pn be properties of graphs. A graph G is (P1,P2, . . . ,Pn)-partitionable (G has property P1◦P2◦ · · · ◦Pn) if the vertex set V (G) of G can be partitioned into n sets V1, V2, . . . , Vn such that the subgraph G[Vi] of G induced by Vi belongs to Pi; i = 1, 2, . . . , n. A property R is said to be reducible if there exist properties P1 and P2 such that R = P1◦P2; otherwise the property R is irreducible. We prove that every additive and inducedhereditary property is uniquely factorizable into irreducible factors. Moreover the unique factorization implies the existence of uniquely (P1,P2, . . . ,Pn)-partitionable graphs for any irreducible properties P1,P2, . . . ,Pn.