Minimal forbidden subgraphs of reducible graph properties

A property of graphs is any class of graphs closed under isomorphism. Let P1,P2, . . . ,Pn be properties of graphs. A graph G is (P1,P2, . . . ,Pn)-partitionable if the vertex set V (G) can be partitioned into n sets, {V1, V2, . . . , Vn}, such that for each i = 1, 2, . . . , n, the graph G[Vi] ∈ Pi. We write P1◦P2◦ · · · ◦Pn for the property of all graphs which have a (P1,P2, . . . ,Pn)-partition. An additive inducedhereditary propertyR is called reducible if there exist additive inducedhereditary properties P1 and P2 such that R = P1◦P2. Otherwise R is called irreducible. An additive induced-hereditary property P can be defined by its minimal forbidden induced subgraphs: those graphs which are not in P but which satisfy that every proper induced subgraph is in P. We show that every reducible additive inducedhereditary property has infinitely many minimal forbidden induced subgraphs. This result is also seen to be true for reducible additive hereditary properties.