How is graphene cut, given that it is similar in strength and hardness to diamond? I assume you cannot just take a pair of regular scissors to a sheet of it and cut out the shape you want?
The word "strength" actually means a lot of different things in different contexts. Some people will use it (incorrectly) to be synonymous with "stiffness." Sometimes it means the ability to resist permanent deformation when subjected to shearing forces ("yield strength" or "shear strength"). Sometimes it means the ability to hold together even when you're trying to pull it apart ("tensile strength"). And these depend on exactly how you define it, and how quickly you apply the forces, the size/shape/microstructure of the material,etc. It's actually a really complicated thing. "Strength" is not a single well-defined quantity for a material. Similarly, there's more than one definition of "hardness."
One thing I can tell you, though, is this: By any measure, just because material A is "stronger" than material B, that doesn't mean you can never use material B to cut material A.
Take a serrated plastic knife, like you'd get out of a picnic-ware set. Could you cut through a 1-cm thick sheet of aluminum with it? Surely not; you'd destroy the knife pretty quickly. Could you cut through a single sheet of aluminum foil with it? Sure. Easily.
The thickness matters!
Steel knives and scissors can easily cut through a 1-atom-thick layer of anything. If you chop through some graphite pencil lead with a razor blade you'll surely cut through lots of layers of, in effect, graphene. Some of the cutting will go neatly between pre-existing layers and crystalline grains and thus not have to actually cut through any layers, but a lot of it won't.
Graphene is only atomically thin, so its strength is not as strong as you imagined. Also, it is a 2D structure with sp2 bonding, which is much weaker than the sp3 bonding in diamonds. For graphene sheets grown on copper foil by chemical vapor deposition (CVD), one can use a scissor to cut off a piece of graphene attached on the copper foil. Then use some etchants to etch away the copper. A CVD graphene then can be deposited on a chip.
To "cut" the graphene at nanometer scales, defining the desired area of graphene by the e-beam lithography followed by plasma ashing is a common method.
These answers do no fully explain the amazing properties of graphene and comparison to graphite is just obtuse. Even though graphene in sheet form is only one atom thick, it has unbelievable sheer strength. The analogy with aluminum foil is not good because it’s sheer strength is very weak in comparison. I doubt you could tear it as easily as sheet foil! The sheer strength of graphene is actually stronger than diamond. As a matter of fact, 2 layers of it can effectively stop a bullet as well as kevlar bullet proof vest.
“Defect-free, monolayer graphene is considered to be the strongest material ever tested, since Hone and coworkers measured the intrinsic strength of the monolayer membrane to be 42 N m−1, which equates to an intrinsic strength of 130 GPa.“