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 PP and QQ be two hereditary graph classes such that PP is closed with respect to the operation of join and QQ is closed with respect to the operation of disjoint union. Let MM be a module of graph GG with associated partition (A,B,M)(A,B,M), where A∼MA∼M and B⁄∼MB⁄∼M, such that G[A]∈PG[A]∈P, G[B]∈QG[B]∈Q and G[M]G[M] is not (P,Q)(P,Q)-split. Then the graph GG is reconstructible.