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Do Yahoo and Google recgonize (or will they delete a listing re-directing) url re-directing?
Depends on what type of redirecting you are referring to. Is this a 301 or 302? Is it a META Refresh?
Also, do they update their search so that the new web site url will appear?
In most instances I would say yes. There are times though when improper redirect methods are implemented and what you thought it was doing, it was really not doing. ;)
Always check your server headers and confirm that the proper http status code(s) is/are being returned from the server.