Is a single node graph a strongly connected component? Can a single node graph be considered a (strongly) connected component?
I'm confused because I was reading about cut vertex which by definition is a vertex that if eliminated increases the number of connected components. 

So if we were to remove vertex number 2, we would get 2 connected components?
Thanks!

A graph is connected if every pair of vertices in the graph can be connected by a path. By definition a single vertex is connected to itself by the trivial path. Hence, it is connected. If the graph is directed the same argument applies. 
See also http://mathworld.wolfram.com/ConnectedGraph.html, specifically:

A graph which is connected in the sense of a topological space, i.e.,
  there is a path from any point to any other point in the graph. A
  graph that is not connected is said to be disconnected. This
  definition means that the null graph and singleton graph are
  considered connected, while empty graphs on n>=2 nodes are
  disconnected.

