# Asymptotics of list size in Robertson-Seymour theorem

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 $g$?

No, it is not linear in the genus; it is at least exponential in $g$. See for example this answer by David Eppstein.
• As Eppstein mentions, if you take a graph which consists of exactly $g+1$ blocks, each of which is a $K_5$ or a $K_{3,3}$, then this will be an excluded minor for embedding in a surface of genus $g$. There are obviously at least $2^{g+1}$ such graphs. Jan 14 '16 at 15:43
It seems that this paper by Djidjev and Reif establishes an upper bound of $\exp(O(g)!)$ for the number of minimal forbidden minors.