Minor and subdivision

Question

It is a well known fact that if $H$ is a graph of maximum degree 3, then $H$ is a minor of a graph $G$ if and only if $H$ is a topological minor of $G$.

Moreover, a graph $G$ has one of $K_{3,3}$ or $K_5$ as minor if and only if it has one of $K_{3,3}$ or $K_5$ as topological minor.

Furthermore, in a writeup by Bruce Reed i saw the following theorem.

Theorem: For any graph $H$, there is a finite set $\mathcal{Z}(H)$ of graphs such that $G$ has $H$ as a minor if and only if $G$ contains a subdivision of some element of $\mathcal{Z}(H)$.

So, my question is under what condition $\mathcal{Z}(H)=\{H\}$ or in other words, what properties must $H$ have so that $\mathcal{Z}(H)=\{H\}$ ?

Answer 1

$\mathcal{Z}(H)$ is the set of graphs obtained from $H$ by splitting vertices of degree $>3$ (the reverse operation to contracting an edge between two vertices, both of degree $\ge 3$, and where the contracted edge is not in any triangle). So (at least when all graphs are assumed to be finite) $\mathcal{Z}(H)=\{H\}$ iff $H$ has maximum degree 3.