A legendary method of executing a mob-hit wherein the assassin slits the throat of his victim and then reaches up inside the mouth and pulls the tongue back out the slit and lets it hang there like a cravat--to the horror of the unfortunate soul who discovers the recently-deceased. Alternatively, "necktie" may refer to a hangman's noose.
Most-famously referenced in AC/DC's "Dirty Deeds":
Done dirt cheap
Done dirt cheap!"