There's one thing worth mentioning here. There's an algorithm that checks if a graph has given minor. If you fix a minor once and for all, the algorithm that ch

There's one thing worth mentioning here. There's an algorithm that checks if a graph has given minor. If you fix a minor once and for all, the algorithm that checks if graph contains a fixed minor has polynomial complexity. Together with Robertson-Seymour theorem, it implies that for any family closed upon taking minors, the problem of determining whether given graph belongs to it, is solvable in polynomial time: just check if it contains forbidden minors. Each minor can be tested in polynomial time, and there are finitely of them, so the whole algorithm works in polynomial time. Now, the thing is: since we don't know the minors, we don't know the algorithm. In other words, we just non-constructively proved that some problem can be solved in polynomial time. We only proved that algorithm exists, we just don't know what it is.