Let graph G have p points v_i and graph H have p points u_i, where p>=3. Then if for each i, the subgraphs G_i=G-v_i and H_i=H-u_i are isomorphic, then the graphs G and H are isomorphic.
It is difficult to prove because the sequences in which corresponding vertices are removed from both graphs usually does not coincide, but finding 'the correct' pair of interchangeable vertices to remove at a definite time is a problem on its own.