TITLE: Mapping graphs to ordinals
QUESTION [7 upvotes]: Robertson-Seymour theorem implies that graph minor relation is a well-quasi-ordering, which means (among other things) that this relation can be extended to a well-order, and other result says that this order can have order type $\psi(\Omega_\omega)$, if I remember correctly. I've been wondering - has anyone actually analysed this ordering? For example, in such ordering, what place would $K_{3,3}$ take? $K_5$? Petersen graph? Only such analysis I know of is on this page: 
http://googology.wikia.com/wiki/User_blog:Hyp_cos/SCG(n)_and_some_related
but it only concerns subcubic graphs, because of their connection to SCG function. 
Thanks in advance for any help!
(If you think this fits better to Math StackExchange, please move the question)

REPLY [2 votes]: There are now results bounding the order type of the set of finite graphs when ordered by the graph minor relation, which is bounded between $\psi_0(\Omega_\omega)$ and $\psi_0(\Omega_\omega^{\omega^\omega})$ with respect to (presumably) Buchholz's ordinal collapsing function. In Rathjen and Krombholz's paper "Upper bounds on the graph minor theorem", this bound is cited from unpublished research, with suggestions for possible future improvements of the lower bound by relating the problem to a generalization of Kruskal's problem of well-ordering trees.