Bipartite operator decomposition of graphs and the reconstruction conjecture

We consider a binary operation • on the set of triads (G,A,B), where G is a graph and (A,B) is the partition of the set of the set V(G). The operation • of multiplication of a triad and a graph is defined and the properties of the introduced operations are described. We study in detail the subcase, when the triads have the form (G,A,B), where G is a bipartite graph and (A,B) is its bipartition. For this case the decomposition theorem stating that any graph except the described family of exceptions can be uniquely decomposed into indecomposable components with respect to the operation • is proved. Using this theorem, we proved the following. Let the graph G have a module M with associated partition (A,B,M), where A∼M and B≁M, such that G[A∪B] is a bipartite graph with bipartition (A,B). Then the graph G is reconstructible.