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In theory you should be able to turn $\ce{CO2}$ to $\ce{CO}$ and then to $\ce{C}$ and connect all the C-atoms together to form graphene.
That should be possible, because plants do something similar, they take $\ce{H2O}$ and $\ce{CO2}$ and link the C-atoms together as $\ce{C6H12O6}$.
Is it possible to produce graphene out of air? Here is a related article from the RCS: "Conjuring graphene oxide from thin air"
A graphene sheet is easily obtained from a block of graphite using adhesive tape. Since graphite itself is essentially a pure grade of coal, it is the result of biological processes in vegetation some millions of years ago. As is pointed out in answers to this Biology.StackExchange question, the majority of a plant's carbon comes from the air via photosynthesis.
So in a sense, we already do that, but we just use the huge stores of graphite from millions of years of high pressure processing beneath our feet rather than start from the beginning.
Yes, any compound of carbon that can be synthesised, can be synthesised from carbon that comes from captured atmospheric $\ce{CO2}$.
We discussed related subjects recently on the new Sustainability Stack Exchange, on the options for plastics without oil.
There are at least three research outfits working on the atmospheric extraction of $\ce{CO2}$. Now, inevitably, capturing it and turning it back into carbon is going to take more energy than gets released from burning coal or gas in the first place to release the $\ce{CO2}$, but that doesn't make the chemistry impossible (and it isn't a deal-breaker economically either, because energy costs vary hugely by time and space.)
And as the article from the UK's Royal Chemistry Society from Jan 2012 that you linked to, highlights, this is already being done:
Graphene oxide can be easily reduced to graphene but is also useful in its own right ... Lee and colleagues produced graphene oxide through a two-step process: first, they fixed the $\ce{CO2}$ to ammonia borane ($\ce{NH3BH3}$) and then they heated the resulting solid to create a honeycomb structure of graphene oxide. In the first reaction, they placed $\ce{NH3BH3}$ into a stainless steel cell and then added $\ce{CO2}$ until the cell reached a target pressure. They then gradually heated the cell to 100°C over a few hours. At a pressure of about 30atm, the mass of the $\ce{NH3BH3}$ doubled thank to the addition of carbon dioxide, producing $\ce{OCH3}$, $\ce{HCOO}$ and aliphatic groups.
Here's the original paper on the research: Formation of Graphene Oxide Nanocomposites from Carbon Dioxide Using Ammonia Borane, Junshe Zhang et al, 2012
There are a few standard ways of making graphene (e.g. exfoliating graphite or chemical vapor deposition), but making it out of air isn't one of these ways (though I suppose the ability to do so would fix a few problems...)
No one has ever been able to do anything like this. I suppose it wouldn't be a thermodynamically favorable process, and the concentration of CO2 in air isn't particularly high.
Graphene can be made from $\ce{CO2}$ by a process of special electrolysis known as SMD. This is a patent pending process which produces graphene in bulk from $\ce{CO2}$ with very little energy. The inventor Michael Nunnerley came across this by accident when he was working on a new system of methane synthesis from $\ce{CO2}$ and $\ce{H2O}$.
