Website was given to someone else, does a "move" or something need to be performed in Webmaster Tools?
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A website was given to another person for their business. None of the original webpages remain they have all been removed. There is nothing on that domain now. Is there some notification that needs to be made in webmaster tools to make Google aware of this?
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Not necessarily. Just remove it from the existing WMT account and ask the new owners to add it in theirs.
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Make sure you do not re-use the old WMT-verification code.
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They will need to take control of the domain themselves, if you have removed all the content and have no involvement in hosting / the domain register anymore you need not worry. The responsability is all on the new owner to validate their new ownership with services like WMT.