Consider the following graph property: for each $u, v \in V(G)$, we have that $G - u \cong G-v$. This property implies a high "symmetry" of the graph.
We can easily see that every vertex-transitive graph has this property. My question: is the converse true?
I've tried to create a counterexample but so far I haven't found any. I appreciate any answer, guidance or reference. Thanks in advance!