For a tree, how to obtain two induced subgraph by breaking an edge in LightGraphs?

Specific Domains Graphs
1

For example, if I have a tree

                  1 - 2 - 3 - 4
                      |     |
                      5     6 - 7 - 8

If breaking the ege (3,6), I want to obtain two induced subgraphs

                  1 - 2 - 3 - 4
                      |   
                      5

and

                  6 - 7 - 8

At present, I implemented a very cumbersome approach, which is

  1. Enumerate the neighbors of vertex 6 (excluding 3 of course) and the neighbors of those neighbors and so on (recursively). Recording the vertices in a list, then serve the list to LightGraphs.induced_subgraph(g, the_list).
  2. Do the same thing for vertex 3.

I wonder if there is a better approach?