Name for relationship where one graph is a minor usually implies another is?

Let $G$ and $H$ be graphs with the following relationship: for some $k$, after you perform at least $k$ arbitrary subdivisions of the edges of $G$ (or the edges produced through subdivision), $H$ must be a minor. What do you call the relationship between $G$ and $H$?

For example, consider a claw with each edge subdivided once and a claw with one edge subdivided twice. You can subdivide any edge of the former to produce a graph for which the latter is a minor $(k = 1)$. Is anyone writing about this or has it been given a name?

• Are you sure you are stating this correctly? Isn't the first paragraph simply "$H$ is a minor of $G$"? Why not add a bunch of isolated vertices, making $k$ irrelevant? – Andrew D. King Jun 13 '11 at 16:51
• Are you looking for topological minor? – Aaron Sterling Jun 13 '11 at 19:27
• @Andrew: you're right, the first paragraph does have a misstatement. I'll edit. – Eli Jun 13 '11 at 19:50
• Is this a correct way to rephrase the relationship you want to describe: "H is a minor of almost all subdivisions of G," where "almost all" means "all but finitely many." – Robin Kothari Jun 13 '11 at 23:08
• @Robin, I think the condition is weaker: after fixing some $k$, after he construct the upward set, he's requiring that $H$ be a minor, for all such sets. – Suresh Venkat Jun 14 '11 at 11:26