#### Vol. 9, No. 4, 2016

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Graphs on 21 edges that are not 2-apex

### Jamison Barsotti and Thomas W. Mattman

Vol. 9 (2016), No. 4, 591–621
##### 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”.

##### Keywords
spatial graphs, intrinsic knotting, apex graphs, forbidden minors
##### Mathematical Subject Classification 2010
Primary: 05C10
Secondary: 57M15, 57M25