degree $2$. Since for every $v$ there are only finitely many such trees, we obtain a finite set of graphs containing $S^+$.) It is easy to see that $\{K_5,\,K_{3,3}\}^+=\{K_5,\,K_{3,3}\}$, but $\Pi^+\ne\Pi$ where $\Pi$ is the Petersen family.

Y. Colin de Verdiere introduced a combinatorially defined invariant $\mu(G)$ of the graph $G$, for which it is known that $\mu(G)\le 3$ iff $|G|$ is planar, and $\mu(G)\le 4$ iff $|G|$ admits a linkless embedding in $S^3$ (see [@LS], [@Iz]).

Van der Holst conjectured that $\mu(G)\le 5$ iff $|\hat G|$ has zero $\bmod 2$ van Kampen obstruction, where $\hat G$ is the cell complex obtained by glueing up all circuits in $G$ by $2$-cells [@Ho]. His supporting evidence for this conjecture is that if $G$ is any of the 78 graphs (called the [*Heawood family*]{} in [@Ho]), related by a sequence of $\nabla$Y- and Y$\nabla$-exchanges to $K_7$ or $K_{3,3,1,1}$, then $\mu(G)=6$, and $|\hat G|$ has nonzero $\bmod 2$ van Kampen obstruction (so in particular does not embed in $S^4$); but if $H$ is a proper minor of $G$, then $\mu(G)\le 5$ and $|\hat H|$ has zero $\bmod 2$ van Kampen obstruction [@Ho].

On the other hand, since $K_7$ contains circuits of length $\ge 