There is a pretty big hurdle to overcome first:
Not only salt, but other dissolved minerals will crystallize on those “precisely sized holes” and clog them up. I suspect they’ll have to abandon the spongelike system they describe here and go with a thin film solution instead, something treated to make it reasonably easy to remove calcium deposits and everything else you’re getting out of the water.
Perhaps there’s another process to separate them? Reduce the graphite back to ash, for instance?
Maybe just wash with more saltwater - it is not a saturated solution (usually) so you can end up with a reasonably cleaned graphite and a bit saltier saltwater.
There’s no requirement to even go beyond the solubility limits. Just distill off clean water, and when the feedstock saline is just about to start dropping crystals, pour it away and get a new, less salty water.
You can potentially even flow the saltwater through the assembly, ending up with a stream of clean water and a stream of slightly saltier water.
And if the combination of graphite cost and lifetime is good enough, it won’t even need to be regenerated.
…thought… could the material be enhanced with some specific ion-binding materials, so the spent graphite becomes e.g. an uranium or gold ore?
Getting the salt out isn’t so bad, but getting calcium crystals off will likely be a nightmare.
Calcium what? Carbonate? Acid helps here.
I wonder how did they make a graphite object large enough (before putting it in the microwave) to insert in a glass and cover the whole water surface.
Sounds like they’ve reinvented the ion exchange resin.
It purifies water, but the exchange ion sites in the matrix fill up. You can elute the ions with acid, but then you’re using chemicals again. And chemicals require expensive purification steps to make …
Purification steps like the removal of ions from water 
who would have guessed that the first stage in water purification would be to plunk a number 2 in it. 
Except that they work by a quite different mechanism. The graphite works by concentrating heat and evaporating water, ionexes work by catching ions to polar materials, typically carboxylated or sulfonated resins.
But yes, there is a certain, albeit fairly distant, similarity.
Why remove it? It’s not as if microwaved graphite is very expensive. Find a constructive use for salty graphite and pour in fresh.
There is a TED talk about all the minerals that are “mined” from the desalination process.
So when there is a certain amount of waster someone can find a use for it. Kind of like the by-products of making charcoal - coal tar -> tar for waterproofing, which can be further refined into kerosene etc.
Yeah, maybe a different mechanism. Also, they bind an indicator dye to the resin so you can tell when it is used up and chuck the cartridge in the trash. With the graphite, it just stays black but quits working. It’s worth noting that there may be some ion-binding action in the little voids in that graphite sponge, you know. Oxidation of the graphite to get carboxyl groups, or sulfur impurities to get sulfonates.
In any case, the laws of thermodynamics still apply. While it’s likely possible to improve the efficiency over boiling water and condensing it back out, you always make a bigger mess someplace else when you purify one thing. Regardless of whether that thing is water or gold or silicon.
That’s what some downstream monitoring is for. In this case it can be as easy as a conductivity probe.
I would dare to say that such effects, while probably present, won’t have any practical significance (unless the material is intentionally functionalized to significant degree). Graphite won’t oxidize much on its own; you can make graphite oxide, which is a pretty useful material, but the prevalent method employs some rather harsh oxidizing agents.
That’s true. If the regeneration of the carbon device is sufficiently easy, e.g. let it soak in the ocean overnight, it’s however not an issue.
Graphite is relatively cheap. Go to graphitestore .com (the first place google threw up) to get sheets for about $30/square meter.
Graphene (a single molecular layer of graphite) is the hypeful material that will solve all of the world’s problems if we can find a way to cheaply make pieces large enough to cover glass of water.
Maybe what we need to do is get the graphite off the calcium crystals?
Most commercial RO machines have backflow, i.e. a cleaning cycle, to clean the pores of the filter.
Asphalt: roads that salt themselves.
certainly but graphitestore.com is not exactly the DIY approach of getting for example a kilo of pencils to shave and have fun while doing so 
If we could easily recover the calcium carbonate, we’d be removing greenhouse gasses from the carbon cycle.
