>>9
I'm not the OP, but usually, no matter what physical condition a part is in, if it works at all, chances are that it's totally fine and stable, as long as it stays cool. (One notable exception is with the electrolytic capacitors on the mainboard)
most of the time when dealing with digital equipment, something either works totally fine or not at all. Besides, if the computer people told you something had melted, they probably replaced it.
That being said, it is most likely a cooling problem, and it is most likely connected to whatever previous incident caused the graphics card to melt. Next time it reboots, feel around inside there and see if either your CPU, GPU, or any other chip in there feels really hot almost hot enough to cause burns on your skin. (obviously, you should cut the power first; I'm not getting sued if you turn out to be a non-functional retard who touches the leads) Whatever fits this description is likely your culprit.
If nothing feels especially hot, it's probably a power issue... and that can be on your mainboard or your power supply.