A graph $G$ is said to be edge transitive provided that, for any two edges $e$ and $f$ in $G$, there is an automorphism of $G$ sending $e$ to $f$.
Call a graph good if it satisfies the following conditions:
- $G$ is connected.
- $G$ is edge transitive.
- For any edge $uv$, the graph $G - \{u,v\}$ is edge transitive.
(By $G - \{u, v\}$, I mean the graph $G$ with vertices $u$ and $v$ removed, as well as all edges incident to either $u$ or $v$.)
Any complete or complete bipartite graph is good, since the removal of any two vertices results in a smaller complete or complete bipartite graph, respectively, which are edge transitive.
The cycle on five vertices is also good, but larger cycles are not. I consider this to be sporadic instance and is not particularly useful for my purposes (though I would still be interested to hear about others).
I think a statement like "A graph is good if and only if it is complete, complete bipartite, or one of these finitely-many graphs" is true, hopefully with a very small "finitely-many". I would be interested in thoughts toward a proof or other examples of good graphs.