I don't want my profile to appear in search results for user profiles.
Is it possible to configure my account to be unsearchable?
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On the Privacy tab in your settings, you'll find a number of settings, however there's no way to become completely incognito. If that's what you want, you'll need to leave Facebook.
Some things you might do to increase your privacy:
But if someone knows your name, they can search for it in Facebook. All you can do is hide everything to non-Friends. You can't hide your name, your profile photo, or your cover photo, however.
You might try a pseudonym, but you may run afoul of Facebook's "real names" policy.