Suppose \(G\) and \(G’\) are graphs on the same vertex set \(V\) such that for each \(v \in V\) there is an isomorphism \(\theta_x\) of \(G-v\) to \(G’-v\). We prove in this paper that if there is a vertex \(x \in V\) and an automorphism \(\alpha\) of \(G-x\) such that \(\theta_x\) agrees with \(\alpha\) on all except for at most three vertices of \(V-x\), then \(G\) is isomorphic to \(G’\). As a corollary we prove that if a graph \(G\) has a vertex which is contained in at most three bad pairs, then \(G\) is reconstructible. Here a pair of vertices \(x,y\) of a graph \(G\) is called a bad pair if there exist \(u,v \in V(G)\) such that \(\{u,v\} \neq \{x,y\}\) and \(G-\{x,y\}\) is isomorphic to \(G-\{u,v\}\).