Is this the indication of failing graphics card?
It could be, yes. The reboot is most likely a kernel panic due to the failing GPU. This, along with the screen distortion, constitute classic signs of failure. It's not the only possible diagnosis though. Though unlikely, it could still be that your GPU is fine and the Display is broken.
How do I diagnose?
As soon as you log in, take a screen shot. If the distortion appears on your screenshot, then the problem is with your graphics processor and you to get your logic board replaced. If the distortion does not appear on the screenshot, then the issue is with your display and you need to get your display replaced.
How do I resolve?
This is either a graphics processor issue or a display issue. I highly suspect the former. Either way, it is a hardware issue for which no online troubleshoot exists. Since this machine is well out of the warranty (and AppleCare) timeframe, it would cost $500-1000 to fix it directly through Apple if you elect to do so. Best of luck.