Some results on decomposable and reducible graph properties

A property of graphs is a non-empty isomorphism-closed class of simple graphs. If P1, ... , P-n are properties of graphs, the property P-1 o ... o P-n is the class of all graphs that have a vertex partition {V-1, ... , V-n} such that G[V-i] is an element of P-i for i = 1, ... , n. The property P-1 circle plus ... circle plus P-n is the class of all graphs that have an edge partition {E-1, ... , E-n} such that G vertical bar E-i vertical bar is an element of 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 is an element of K such that P = P-1 o P-2 . P is decomposable over K if P = P-1 circle plus 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? (C) 2012 Elsevier B.V. All rights reserved.