One-to-one vs one-to-zero Relationship

Question

So we have like two relationships one is one-to-one and the other is one-to-zero. And let me explain what I mean by that. By one-to-one I mean that there is no parent, one row cannot be identified without the other. By one-to-zero I mean that parent can exist whenever it wants but the child needs parent to identify itself. The question is how can I make a one-to-one relationship with MySql using InnoDB?