This is a questions I've been playing with the last few days:
Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. We will denote the vertex-deleted induced subgraph obtained by deleting $v\in V$ as $G_v$. Given two vertices $x,y\in V$ such that $G_x\cong G_y$, when does an isomorphism $\phi:G_x\rightarrow G_y$ extend to an automorphism, $\tilde{\phi}:G\rightarrow G$?
First thought: it would seem that the structure of such a graph would have some sort of symmetry described by and action of $\mathbb{Z}_2$ on $\{x,y\}\subset V$ (i.e. there is a map $\psi(x)=y$ that extends to an automorphism on $G$). This is based on intuition an may be completely wrong.