copy of the triod, identified with the remaining edges of the $1$-skeleton of $\Delta^3$.

Of course, (a) can be reformulated in terms of minors, for given a minor $H$ of $G$ and a knotless and linkless embedding of $|G|$ in $S^3$, the homeomorphism $S^3/\text{tree}\cong S^3$ (see Lemma \[trivial\]) yields a knotless and linkless embedding of $|H|$ in $S^3$, unique up to equivalence (by an orientation-preserving homeomorphism of $S^3$).

Robertson and Seymour have proved, in a series of twenty papers spanning over 500 pages, that every minor-closed family of graphs is characterized by a finite set of forbidden minors (see [@Di] for an outline). Let us abbreviate “a subdivision of a subgraph” to a [*$\tau$-subgraph*]{}. There exists a $\tau$-subgraph-closed family of trees that is not characterized by a finite set of forbidden $\tau$-subgraphs (cf. [@Di §12, Exercise 5]).

However, it is well-known and easy to see that every minor-closed family of graphs that is characterized by a finite set $S$ of forbidden minors is also characterized by a possibly larger but still finite set $S^+$ of forbidden $\tau$-subgraphs. (For each $G\in S$ and each $v\in G$, replace the star $\operatorname{st}(v,G)$ by a tree with $\deg v$ leaves and no vertices of 