You are given a very special graph. The vertices of the graph come in three columns: left, center, and right. The edges connect vertices from the left to vertices in the center, and from the center to the right; there are no left to right edges. The graph also has the property that it does not contain any “closed diamonds”, meaning there are never two routes from a vertex on the left to a vertex on the right. See sample graphs

You are asked to prune the graph following these rules. For each left vertex, remove all but one of its outgoing edges. When you remove a (left,center) edge, then also remove the corresponding center vertex. When you remove a center vertex, then also remove all of its (center,right) edges and the corresponding right vertices.

Is it possible in this way to remove all of the right vertices? If so, give an example. If not, prove it. (I believe it is not possible, and I’m looking for a proof.)