Additive and hereditary properties of graphs are uniquely factorizable into irreducible factors

A hereditary property of graphs is any class of graphs closed under isomorphism and subgraphs. Let P1,P2, . . . ,Pn be hereditary properties of graphs. We say that a graph G has property P1◦P2◦ · · · ◦Pn if the vertex set of G can be partitioned into n sets V1, V2, . . . , Vn such that the subgraph of G induced by Vi belongs to Pi; i = 1, 2, . . . , n. A hereditary property is said to be reducible if there exist hereditary properties P1 and P2 such that R = P1◦P2; otherwise it is irreducible. We prove that the factorization of a reducible hereditary property into irreducible factors is unique whenever the property is additive, i.e., it is closed under the disjoint union of graphs. c © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 44–53, 2000