Question on large induced subgraphs of non-isomorphic graphs

Question

Let $G$ and $H$ be non-isomorphic graphs and consider a bijection $f:V(G)\rightarrow V(H)$. Is it necessarily true that there is some vertex $v\in V(G)$ for which the subgraphs induced by $V(G)\backslash \{v\}$ and $V(H)\backslash \{f(v)\}$ are not isomorphic?

Answer 1

No. Consider $G= K_2$ and $H$ the complement of $G$, i.e., $H$ is two isolated vertices. Then, for any bijection $f:V(G)\to V(H)$, and any $v\in V(G)$, the subgraphs induced by $V(G)\backslash \{v\}$ and by $V(H)\backslash \{f(v)\}$ are both just a single vertex, and thus isomorphic.