Some results on decomposable and reducible graph properties

A property of graphs is a non-empty isomorphism-closed class of simple graphs. If P 1 , … , P n are properties of graphs, the property P 1 ∘ ⋯ ∘ P n is the class of all graphs that have a vertex partition { V 1 , … , V n } such that G [ V i ] ∈ P i for i = 1 , … , n . The property P 1 ⊕ ⋯ ⊕ P n is the class of all graphs that have an edge partition { E 1 , … , E n } such that G [ E i ] ∈ P i 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 P 1 , P 2 ∈ K such that P = P 1 ∘ P 2 . P is decomposable over K if P = P 1 ⊕ P 2 . We study questions of the form: If P is reducible (decomposable) over K 1 , does it follow that P is reducibe (decomposable) over K 2 ?