Minor and subdivision
$\mathcal{Z}(H)$ is the set of graphs obtained from $H$ by splitting vertices of degree $&gt;3$ (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.