Hello.
Suppose we have a undirected graph V. We certainly know that any vertex u can be reached from any vertex v.
If we delete vertex K and it's edges and, therefore, any vertex u losses connection with any vertex v, then K is considered to be important.
So we need to find the number of this important vertexes.
First that comes to mind is DFS, but I think there are other faster ways to solve this problem.
This is just the number of critical vertexes in a graph. The idea is to use Trajan's algorithm.
Here is a link to Geek for Geeks tutorial about that.
http://www.geeksforgeeks.org/articulation-points-or-cut-vertices-in-a-graph/