Reconstruction from vertex-switching

Let X be a graph with vertices x₁ ,…, x_n. Let X_i be the graph obtained by removing all edges {x_i, x_j} of X and inserting all nonedges {x_i, x_k}. If n ? 0 (mod 4), then X can be uniquely reconstructed from the unlabeled graphs X₁.…, X_n. If n = 4 the result is false, while for n = 4m≥8 the result remains open. The proof uses linear algebra and does not explicitly describe the reconstructed graph X.