TITLE: How unhelpful is graph minors theorem?
QUESTION [7 upvotes]: A very interesting Robertson-Seymour (graphs minors) theorem says:

Any infinite collection of graphs $C$ with the property that if $G\in C $ then its minors also are has the form $\{$graphs $G$ that don't contain any $E_i\}$ for some finite collection $E = \{E_i\}$.

So, the theorem says that you could create a list of forbidden minors to find out if the graph is torically embeddable, but this doesn't help much, since the list is both not fully known and large.
I wonder whether the above difficulty is because 

it is indeed hard to test this property of a graph
the theorem does turn easily testable properties into long lists
it is not known how to effectively turn easily testable properties into lists

Here's the formal question:

Consider a polynomial algorithm $P$ that returns a yes/no question given a graph as an input and which always returns yes for minors of any graph for which it returns yes. There exists $E$, the exceptional list of a collection defined by $P$. What is known about the computability of the map $P\mapsto E$?

REPLY [3 votes]: To answer some of your questions:
1. Yes, testing such properties is usually hard. For instance, before the graph minors theorem we had properties for which no algorithm was known at all! (Maybe a recursively enumerable algorithm was known, I don't remember.) After the graph-minors theorem, such properties became testable in polynomial time! Now that's a big jump from no algorithm known to polynomial time. (If I remember correctly, the polynomial is like O(n log n), which is almost linear time.)
As for 2. and 3., I don't know any property which we can easily test, for which we don't know the forbidden minors. I'd like to know such properties, if they are known. It seems to me that if we have an algorithm for easily testing a property, we really understand the property, and therefore should be able to come up with a list of forbidden minors somehow. Of course, this is just a feeling.
EDIT: I've been corrected in the comments. Please read Gil Kalai and David Eppstein's comments.