A planar graph cannot have $K\_5$ and $K\_{3,3}$ as minors. Robertson-Seymour theorem generalizes this by stating for every genus $g$ there is a finite list of forbidden minor graphs that are obstructions that prevent the graph from being genus $g$. Is there any result on the size of the list? Is it linear in $g$? 



Asymptotics of list size in Robertson-Seymour theorem
No, it is not linear in the genus; it is at least exponential in $g$. See for example this [answer](https://cstheory.stackexchange.com/questions/8876/forbidden-minors-for-bounded-genus-graphs) by David Eppstein. 


