Why is my MBA restarting on its own?

Help!

My MacBook Air is restarting on its own (typically after about a minute or so, and it takes something on the order of five minutes before it comes back on).

Come to think of it, I've been using workarounds for a while -- I haven't been able to connect it to an external monitor for a year or so or else it would shut itself down (seemed like it would get a bit warm in the center rear of the base). Now it's doing that even if I run it without connecting to an external monitor.

Doesn't matter whether I'm on battery power or plugged in. It may be load dependent since if I turned it on and didn't do anything it seemed to stay on longer, but if I turned it on and, say, tried desperately to copy off my data to a thumb drive, it would turn off within a minute.

Took it to the Apple Store and it started up with a noticeable visual disturbance on the screen (a couple of horizontal lines flickering at one point in the startup scheme) but wouldn't run long enough for them to complete the diagnostic before cycling power, so they told me they think it's a logic board problem.

Since my issues started with an external monitor that makes me think graphics, so is the graphics integrated on the logic board for this model (A1465/EMC 2558)?

How to fix?

I've already opened up the case and pulled the SSD to directly retrieve my data...