Graphs G and H are hypomorphic if there is a bijection $ϕ$ : V(G)$\to V(H)$ such that G-u$\cong H-ϕ(u)$, for all $u\in V(G)$. The reconstruction conjecture states that hypomorphic graphs are isomorphic, if G has at least three vertices. We investigate properties of the isomorphisms G-u$\cong H-ϕ(u)$, and their relation to the reconstructibility of G.