Well, I know in the desktop graphics market, ATI have won, hands down. But not so in the mobile market (and we have to include this when we're generalising like this). NVIDIA have Optimus technology, and it's forcing a lot of laptop manufacturers to choose NVIDIA. People want the option of a powerful graphics card on tap but the option of reverting to integrated when increased battery life is required.
Yes, traditional switchable graphics is an option for both but due to increased costs and complexity (it requires extra mulitplexers/routing on the mainboard) many mobile manufacturers have opted not to go down the "traditional route" and choose NVIDIA instead. It was also proved that with traditional SG, users rarely manually switched and left discrete GPU on all the time, drastically reducing battery life.
So, it would be unfair to say that ATI have won completely. The majority of Asus's new machines utilitse Optimus (w/ NVIDIA discrete GPU's) and the new MacBooks are also rumoured to use Optimus also. With no real mention of NVIDIA releasing DX11 mobile GPU's anytime soon, and if ATI can develop a software similar to Optimus over the next couple of months, then I would say without a doubt, that they have completely conquered the computer graphics market.