I'm reading On random graphs by Erdos and Renyi and they define the completely connected graph as the graph that effectively contains all vertices $P_1,\dots P_n$ (has no isolated points) and is connected in the ordinary sense.
I dont see how being completely connected is stronger than being connected in the ordinary sense. Do they not mean
"and is connected in the ordinary sense"
as
"and in addition, has the property of being connected in the ordinary sense".
?
In other words: is there a (classically) connected graph which is not completely connected (i.e. which does not contain all points)? Or maybe that was the definition of connected back then: being one connected component plus some single vertices?