I have a graph which is a tree actually.
I need to delete the whole subtree given a vertex.
Is there any interface to delete the whole subtree given the vertex ?

or else, I will have to write a routine which does DFS and deletes
each vertex/edge.

Correct me if I am wrong:
If I do
clear_vertex(vertex_to_be_deleted, g)
remove_vertex(vertex_to_be_deleted, g)

I wont achieve what I want and I will get 2 disconnected graphs ?

Generally, yes. Unfortunately, subgraph doesn't implement remove_vertex. It's slated as under construction, but appears to have been that way for quite some time.