Factorizations of properties of graphs

A property of graphs is any isomorphism closed class of simple graphs. For given properties of graphs P1,P2, . . . ,Pn a vertex (P1,P2, . . . ,Pn)-partition of a graph G is a partition {V1, V2, . . . , Vn} of V (G) such that for each i = 1, 2, . . . , n the induced subgraph G[Vi] has property Pi. The class of all graphs having a vertex (P1,P2, . . . ,Pn)partition is denoted by P1◦P2◦ · · · ◦Pn. A property R is said to be reducible with respect to a lattice of properties of graphs L if there are n ≥ 2 properties P1,P2, . . . ,Pn ∈ L such that R =P1◦P2◦ · · · ◦Pn; otherwise R is irreducible in L. We study the structure of different lattices of properties of graphs and we prove that in these lattices every reducible property of graphs has a finite factorization into irreducible properties.