Some results on decomposable and reducible graph properties

A property of graphs is a non-empty isomorphism-closed class of simple graphs. If P1,…,Pn are properties of graphs, the property P1∘⋯∘Pn is the class of all graphs that have a vertex partition {V1,…,Vn} such that G[Vi]∈Pi for i=1,…,n. The property P1⊕⋯⊕Pn is the class of all graphs that have an edge partition {E1,…,En} such that G[Ei]∈Pi for i=1,…,n. A property P which is not the class of all graphs is said to be reducible over a set K of properties if there exist properties P1,P2∈K such that P=P1∘P2. P is decomposable over K if P=P1⊕P2. We study questions of the form: If P is reducible (decomposable) over K1, does it follow that P is reducibe (decomposable) over K2?