Testing $H$ a topological minor of $G$ --- algorithms that scale with $H$? I am trying to find about algorithms that, given graph $H, G$, determine if $H$ is a topological minor of $G$ (and if so, exhibit this explicitly). Most of the literature on this topic seems to be focused on the case when $H$ is a fixed small graph while $G$ goes to infinity --- e.g. testing if $G$ is planar. What are the algorithms when $H$ and $G$ are comparable in size? 

If this problem requires exponential time, I would still be interested in the best algorithms and/or real implementations
The general problem when $H$ is not fixed is NP-hard: Take $H$ to be a cycle of length $|V(G)|$ and it amounts to testing whether $G$ is hamiltonian.

I think that the state of the art on this is the FPT algorithm of 
Grohe, Kawarabayashi, Marx and Wollan. The dependency on $G$ is cubic, but the one on $H$ seems impractical, since it relies quite heavily on the Robertson-Seymour theory.