How to remove all trace of SSID & Network Security Key

Having made a partial link to my LAN on a laptop with Vista Home Basic, the
laptop is being given to another and despite the settings I change, I still
seem to be able to relink to the LAN without having to enter any security
details - even though it's secure.

Having entered the details once, I can't seem to remove them permenantly.
Any suggestions would be appreciated (although reformatting is not an option).