Four days later, I was told that the part was still on backorder and should be there by the end of the week. The following day I was told that Apple had informed the repair facility that the part was on backorder and would take another three days to arrive. The logic board, however, had to be ordered from Apple but would be there the following day.
#MACBOOK PRO MID 2010 LOGIC BOARD REPLACEMENT PRO#
So, I took my MacBook Pro to the authorized repair facility and was told that it failed the logic board test and that the logic board would be replaced at no charge under AppleCare. When the "Genius" at the "Genius Bar" says that he has no idea what I'm talking about when I mention an issue raised by thousands of Apple users on hundreds of forums, only to hear from the "genius" that he has never heard of the issue, "It requires a strong moral principle to prevent me from" literally saying out loud to anyone in the range of my voice: "these aren't the droids you're looking for" (my apologies to anyone offended by my conflating "Moby Dick" and "Star Wars"). I hate the "Genius Bar" because I feel that the people at the Genius Bar are either incredibly uninformed or are outright liars.
I took it to an Apple authorized facility rather than the Genius Bar. It had all the classic symptoms of the graphics card issue.
The logic board on the machine I use first failed in May 2013. Though all have been replaced under AppleCare, it is clear that Apple has a quality control problem. I have onen mid 2010 MacBook Pro (Purchased in late 2010) that is on its third logic board since May 2013 (and a failed optical drive), and two other that are each on their second logic boards.