# If the parent in a bridge is the root, is it a bridge cut node?

I am going through Professor Skiena's textbook and he says that there are 3 types of cut-nodes - root, bridge cut node, and parent cut node.

Now if the earliest reachable ancestor from a node is itself, it's parent and the node (if not leaf) become bridge cut nodes.

My question is that we have a simple 2-node undirected graph -> A --- B.

In this case, B's earliest reachable ancestor is B itself and hence A becomes the bridge cut node. However, I am unsure that should be the case; since if I delete A, even then B exists and the num_components = 1?

• Could you give a more precise reference to what you're talking about than "[somewhere in] Professr Skiena's textbook"? – David Richerby Aug 2 '18 at 10:39

By definition, a graph that has only one vertex is connected.
That's to say if you delete A, the graph B is still connected. Clearly, A is not a cut-node because it can't disconnect the graph. So is vertex B.