Operator decomposition of graphs and the reconstruction conjecture

We present the method of proving the reconstructibility of graph classes based on the new type of decomposition of graphs — the operator decomposition. The properties of this decomposition are described. Using this decomposition we prove the following. Let P and Q be two hereditary graph classes such that P is closed with respect to the operation of join and Q is closed with respect to the operation of disjoint union. Let M be a module of graph G with associated partition ( A , B , M ) , where A ∼ M and B ⁄ ∼ M , such that G [ A ] ∈ P , G [ B ] ∈ Q and G [ M ] is not ( P , Q ) -split. Then the graph G is reconstructible.