iMac A1311/A1312 Graphics card
iMac Late 2009 to Late 2011 are known for their GPU failure. All models are affected to a certain extent, except the 21.5" 2009 with integrated graphics only (NVidia 9400M). All other models use a ATi/AMD Radeon HD HD4000 through HD6000 series GPU which are unreliable. The most affected is the Radeon HD6000 series in the 2011 models, then the Radeon HD5000 in the 2010 models and finally the Radeon HD4000 in the 2009 models. Heat does reduce the lifespan of these, so higher-end models are more likely to fail. If you are still using one of these, it is highly recommended put fresh thermal paste and to install Macs Fan Control and set a much more aggressive fan profile. This is also recommended for older models (2006 to 2009) which are also failure-prone.
There are no officially compatible graphics card available to replace them. Most cards you find on eBay or elsewhere already have been baked or will fail soon enough.
The best solution is to install a newer unsupported graphics card. However, Apple's EFI does not support the VBIOS used on the "non-Apple" cards, some people thankfully figured out how to patch the VBIOS to restore EFI video output support as well as other features (native brightness adjustment, external display support) on some cards.
If you are considering replacing the GPU in your iMac, you should REALLY read this post as it is appears to be the central repository for all things related to iMac GPU replacements - including more information on supported cards and available VBIOS: https://forums.macrumors.com/threads/2011-imac-graphics-card-upgrade.1596614/
Note that A1225 2009 and A1311 2009 DG only support some older cards with native LVDS output rather than eDP. These cards from around 2012 may not have the best reliability either. Additionally, they require using OpenCore to boot into macOS. There is no fully transparent solution allowing to boot macOS natively. However, since replacement graphics cards support Metal, it is also an opportunity to install a newer macOS version with OpenCore Legacy patcher and get acceptable performance without graphical glitches. For the A1311 2009 DG, it is possible to swap the logic board with the A1311 2009 IG model to get rid of the discrete graphics card. It is not easily possible to convert an A1311 2009 DG board to IG board since it requires swapping the MCP from "NF-7201-DC-B3" to "GF-9400J-DC-I-B3" and removing the parts with BOM option "MXM" then soldering the parts with BOM option "IG"