To disconnect from a contact click on the green connect icon (which will turn red as you hover over it). The connect icon will turn grey, indicating no connection.
Disconnecting from a contact will instantly remove them from your private graph and vice versa. Your contacts will not be accessible to them and your profile updates will not be visible to them.
The person who you've disconnected from will not receive any notification that you have disconnected. The connect icon for your contact card in their address book will become grey.