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?