Not based on what is known now. Need more clues.
The graphics problem could be pre-symptomatic of a larger problem. Or it could be a stand-alone problem.
The shutting off problem could be a power problem, an overheating problem, or it could be a symptom of the graphics problem.
The graphics problem could be software and/or driver
related, or it could be hardware related. As could the power problem.
A bad battery or power supply has been known to cause your kind of power symptom. You want to try running the computer for a while on battery only, on power supply only (remove the battery from the laptop and run it without one) to test to see if this could be the cause. You said you did this test and it was non conclusive. So we are assuming these components are good.
Overheating has been known to cause both your graphics problem and your power problem. In a laptop this is normally due to:
- A fan has stopped working
- Dust has built up inside the laptop, blocking air flow.
- A Component is overclocked, overworked, or defective and is running hot.
It would be much easier if I had your laptop on my bench. But right now all I see is SevenForums! The more information you can give the more help I can be.
And very often, in the process of slowing down and explaining their problem carefully a lot of folks figure out what is going on themselves.
But for now I suggest exploring the overheating issue. Here are some video tips: How To Clean Dust From Your Laptop - YouTube How to Clean a Laptop Fan - YouTube
Lets see if cleaning does it.