I'm investigating finite, simple graphs with the following property: For each degree $d$ of $G$, the subgraph induced on all vertices of degree $d$ is vertex transitive.
In particular, I'm interested in graphs that are not vertex transitive to begin with.
For example, consider the following graph.
Since it has vertices of degree 4 and 5, it is not vertex transitive. But the subgraph induced on the four vertices of degree 4 is $C_4$, which is vertex transitive. Likewise, the subgraph induced on the four vertices of degree 5 is $K_4$, which is vertex transitive.
The house graph is a nonexample, as the subgraph induced on the three vertices of degree 2 gives $K_1 \cup K_2$, which is not vertex transitive.
My initial questions:
Is there a name for this property?
Is much known about graphs with this property?
I'm thinking this property might be similar to saying that for each degree $d$, all vertices of degree $d$ are in the same orbit under the automorphism group of $G$, but I haven't fleshed that out fully yet.
Thank you.