I have a graph $G(V, E)$. I want to know the number of unique subgraphs with diameter d >= 1. I will give a couple of examples:
1) In the following graph, there is 3 unique subgraphs of diameter 1.
2) In this one, although it contains 3 vertices like the first one, has a single subgraph of diameter 1. (itself)
3) In this one, there is 3 of diameter 1.
Maybe I can explain my problem algorithmically, since I am not a mathematician and researching the problem did not help.
For every vertex $v \in G(V)$, get the subgraph containing $v$ such as the distance from $v$ to every other vertex in the subgraph is less than or equal to $d$. So what I want is not how to get all subgraphs, but the count of subgraphs. (Well if there is an answer for how to get them in an efficient way I don't mind it as a bonus).