Car dies

When slowing down, car will shut down, and die. Thought was low in fuel. No difference. When in traffic. At stoplight. Car will die. Turns over strong, seems like it wants to start, just as I stop turning the key. It may catch, and go, until car slows down again, and again it dies.
mechanic checked with guage the fuel pump. Says its okay.
perhaps fuel regulator, or ingection problem. A relay. Or 02 sensor?
what things should my mechanic check, and in what order? Thanks