Operator decomposition of graphs and the reconstruction conjecture

A new type of decomposition of graphs – the operator decomposition connected with the classical notion of homogeneous set (or module) is introduced. 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 6∼ M , such that G[A] ∈ P , G[B] ∈ Q and G[M ] is not (P, Q)-split. Then the graph G is reconstructible.