Graphs on 21 edges that are not 2-apex

Abstract

We show that the 20-graph Heawood family, obtained by a combination of $\nabla Y$ and $Y\nabla$ moves on $K_7$, is precisely the set of graphs of at most 21 edges that are minor-minimal with respect to the property “not $2$-apex”. As a corollary, this gives a new proof that the 14 graphs obtained by $\nabla Y$ moves on $K_7$ are the minor-minimal intrinsically knotted graphs of 21 or fewer edges. Similarly, we argue that the seven-graph Petersen family, obtained from $K_6$, is the set of graphs of at most 17 edges that are minor-minimal with respect to the property “not apex”.