Publishing only the built .sln files artifacts in TeamCity?

I don't think there is a way to do it out of the box, nor I'm aware of a relevant plugin. However, you can achieve the desired result by comparing new binaries produced by the build with previous ones and publishing only those that differ.

TeamCity provides VCS Trigger Rules which you can use to only trigger builds when files under a certain path within a repo are changed. You can use these rules in conjunction with Checkout Rules which are used to fine tune what is actually pulled down from the repo.

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