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 ℘1, ℘2,…, ℘n be hereditary properties of graphs. We say that a graph G has property ℘1°℘2°…°℘ n 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 ℘i; i = 1, 2,…, n. A hereditary property is said to be reducible if there exist hereditary properties ℘1 and ℘2 such that ℛ = ℘1°℘2; 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. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 4453, 2000