We will continue our discussion on garbage collection. The process has some steps that are not obvious. We will go over them in this post.
First, we look at the propagator step we added.  The propagator tests a condition for each edge. This condition marks an edge between x and y as ok if x is a root and with an edge between x and y, they are connected with y as root. That is the propagator takes each garbage and connects them to food.  It continues execution until a termination condition has been reached where the root is connected with all the vertices visited such that each edge x,y is ok.
Between two edges there is only one condition when the edge is not ok. This is when x is not a root. If x is not a root, it is manure. A manure is now easy to collect.
The fact that x is manure and is not a root is maintained as an invariant with the progress because we add edges. Notice that in the ok condition, we checked and maintained that y is root. This is a refinement from the earlier definition where we considered y is food. This let us come up with two separate connected graph where one is manure and the other is food. Everything else is unconnected and garbage.
To be able to prove that this is the case, we considered our actions and maintain the invariant that if x is manure it is not a root. Said differently, each vertex is neither manure or not a root for the marker to propagate. We show that this invariant holds throughout the progress.
Initially there is a root for food and  manure is part of garbage. This is easy to establish because the food is mutually exclusive. By definition, food is any vertex that is reachable to the root. So our initial condition holds.
Next in our actions, we look at the two that change a vertex from being or not being a root.
The first is when we add the edge between x and y  and y is food. In this case since y is food, it is not manure. Since the initial invariant holds for all vertex that is not y, adding an edge from that vertex to y, maintains the invariant.
The other action we do is when x is a root and now there exists an edge between x and y. In this case we proceed from the left. If x is not manure and we connect y, then y is not manure. if y is not manure, then y is root. This re-establishes our invariant.
Thus by maintaining our invariant and our progress, we are able to propagate. we detect the termination condition when the root is chosen and all the edges reaching to it are ok.