Graphite can be thought of as various layers of graphene mounted on top of each other. Graphene is known to be as robust as diamonds, yet graphite can be found in pencils and we all know from elementary school how fragile they are. I know that this has to do with the graphene layers being bonded weakly by Van der Waals. On the contrary, the carbon atoms have much stronger covalent bonds. Does it mean that when we break a pencil, we're always separating layers? If so, what's up with pencil mine dust?
Yes, the force between the graphene layers is relatively weak. It is a bit like slate, which breaks very easily along is layers, but is quite strong in the XY plane. Modern graphene is also produced in larger sheets.