neither does (one)
One also does not. A: "I really don't like pineapple on pizza." B: "Blech, neither do I." A: "How will you get work? You don't have your high school diploma!" B: "Neither do a lot of billionaires!"
neither does someone
[does] not either. Susan does not own a cat, and neither does Mary. Bill doesn't want to see a movie tonight, and neither do I.