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.