I have a problem from graph theory for which I could need some theoretical background and, if possible, ideas for an algorithm.
Suppose we have a "usual" graph $G =(V,E)$ with finitely many vertices $V$ and edges $E$, at most one edge between two vertices, undirected. We can assume $G$ to be connected, it can, and often will, contain cycles. The graph is known to the algorithm.
Now each edge can have two states: "blocked" or "unblocked". The blocked or unblocked state of an edge is hidden to the algorithm.
Two vertices $v$ and $w$ are "good connected" if they are connected by a sequence of unblocked edges.
One also says "$v$ is reachable from $w$" (and vice versa).
Now assume we have a set $P$ of pairs of vertices $(v, w)$.
The algorithm can, for all vertex pairs $(v,w) \in P$ get the information whether they are good connected or not.
We want to check if for all $(v,w) \in P$ the vertices $v$ and $w$ are good connected. As $(u,w)$ is good connected if $(u, v)$ and $(v, w)$ are good connected there might be redundancy in the set $P$. Other forms of redundancy (for example from splittability of $G$) can also occur.
So we would like to have a "minimal test set" $P' \subseteq P$ of $(v',w')$ so that, if all $(v',w') \in P'$ are good connected, also all $(v,w) \in P$ are good connected. With "minimal" I mean: "Having the minimal number of elements", not "minimal under set inclusion".
Is this a problem already studied? Are there efficient algorithms for it?
