Robertson and Seymour tell us that any minor-closed family of graphs has a finite collection of excluded minors.


Standard examples include planar graphs with two excluded minors ($K\_5$ and $K\_{3,3}$) and knotlessly embeddable graphs with the Petersen family as excluded minors.


However in general, a finite list is not necessarily a short list, and there are natural properties with large numbers of excluded minors. In fact, I recall seeing examples where astronomical numbers of excluded minors have been proved to exist - unfortunately for the life of me, I can't remember these examples and I can't find them on MathSciNet.


So my question is: please give me references to results that show the existence of vast numbers of excluded minors for natural minor-closed classes of graphs.



Minor-closed classes of graphs with large numbers of excluded minors
Here is another answer that I just recently learned about that I find quite shocking. A graph is called $Y \Delta Y$ *reducible* if it can be reduced to isolated vertices by applying any of the following six operations.


1. Delete a loop.
2. Delete a degree one vertex.
3. Replace a pair of parallel edges by a single edge.
4. Replace an induced path by a single edge.
5. $Y \Delta$ transformation: Delete a degree 3 vertex $w$ and its three incident edges $wx,
wy$ and $wz$, and add three edges $xy, yz$ and $xz$.
6. $\Delta Y$ transformation: Delete the three edges of a triangle $\Delta:=xyz$, and add a new vertex $w$ and three new edges $wx, wy$ and $wz.$


Truemper showed that the class of $Y \Delta Y$ reducible graphs is minor-closed. Epifanov proved that all planar graphs are $Y \Delta Y$ reducible. Recently, [Yu](http://www.kurims.kyoto-u.ac.jp/EMIS/journals/EJC/Volume_13/PDF/v13i1r7.pdf) showed that there are more than 68 billion excluded minors for $Y \Delta Y$ reducibility. 


Specically, Yu finds 68897913659 graphs that are excluded minors (up to isomorphism).


