# What is the best upper bound on the running time of the graph minor algorithm?

A cornerstone of the graph minor theory is an algorithm that, given undirected graphs $$G, H$$, runs in time $$f(|H|)poly(|G|)$$, and determines whether $$H$$ is a minor of $$G$$ or not. It has been obtained from the famous result of Roberson and Seymour.

This result, as well as other algorithms for minor testing, is always cited with $$f$$ being some implicit computable function. I wonder what are the best known upper bounds on $$f$$, e.g., is it elementary?

In this discussion, user @Saeed mentions that $$f$$ is at most quadruple-exponential but provides no reference for this claim.

• In this paper, $f$ is described as "recursive" and with an "immense exponential growth". Some references in the paper may help. – Michele Amoretti May 17 at 6:59

• Where in the paper is the bound stated? All I can see is a $2^{2^{2^{2^{\Omega(k)}}}}$ lower bound, but no upper bounds. – Emil Jeřábek May 17 at 10:14