Self Shutting Down Arduino (or any other microcontroller, for that matter)

If a locking relay was attached to a micro-controller's power-supply, couldn't said micro-controller shut itself off? Does a locking relay require a pulse of a certain length? I am an absolute beginner in electronics and I would love improvements/suggestions. I would be sending a command from a computer (pi?) for my robot to shut itself off and save the map of its environment. This led me to thinking about remote turn on. Could some sort of unpowered rf receiver be used to trigger said relay? I would love suggestions/ideas/part numbers for CHEAP rf modules that would work for this.

thanks

-gabriel