Neighborhood Reconstruction and Cancellation of Graphs

We connect two seemingly unrelated problems in graph theory. Any graph G has a neighborhood multiset N (G) = {N(x) | x ∈ V (G)} whose elements are precisely the open vertex-neighborhoods of G. In general there exist non-isomorphic graphs G and H for which N (G) = N (H). The neighborhood reconstruction problem asks the conditions under which G is uniquely reconstructible from its neighborhood multiset, that is, the conditions under which N (G) = N (H) implies G ∼= H. Such a graph is said to be neighborhood-reconstructible. The cancellation problem for the direct product of graphs seeks the conditions under which G×K ∼= H ×K implies G ∼= H. Lovász proved that this is indeed the case if K is not bipartite. A second instance of the cancellation problem asks for conditions on G that assure G ×K ∼= H ×K implies G ∼= H for any bipartite K with E(K) 6= ∅. A graph G for which this is true is called a cancellation graph. We prove that the neighborhood-reconstructible graphs are precisely the cancellation graphs. We also present some new results on cancellation graphs, which have corresponding implications for neighborhood reconstruction. We are particularly interested in the (yet-unsolved) problem of finding a simple structural characterization of cancellation graphs (equivalently, neighborhood-reconstructible graphs).