Help Needed in a Problem

In our Design and Analysis Project, we came across a problem of which we have very little or no idea. The problem essestially breaks down to :

Given a rooted tree, find a minimum dominating set such that no two vertices in the dominating set share an edge and for all the vertices that is not in the dominating set, there exists exactly one vertex in the dominating set which is its neighbor. If such a set does not exists, report -1.

Any help as to how to approach the problem will be really appreciated.
Thank you.

dominating in the sense?