Graph isomorphism of two graphs that have isomorphic subgraphs

Question

Suppose we have two graphs that are isomorphic to each other ($G$ and $H$). We also have the bijection between the vertices of these two graphs. Now we add one edge to each graph ($G+e$, $H+e$). Is there any easy way to find whether the resulting graphs are still isomorphic or not? And also find the bejections among the nodes? I truly appreciate any help.

Answer 1

Suppose we know every isomorphism $\phi$.

If the edge inserted into $G$ is $e = (v_{1}, v_{2})$, then check whether the edge inserted nto $H$ is $e' = (\phi(e_{1}), \phi(e_{2}))$.

This needs to be done for every isomoprhism $\phi$. If the check fails for every isomorphism, then the graphs $(G + e, H + e')$ are not isomorphic.