Graphs on 21 edges that are not 2--apex

TL;DR

This paper characterizes specific minor minimal graphs related to 2--apex and intrinsically knotted properties, focusing on graphs with up to 21 edges derived from well-known graph families.

Contribution

It precisely identifies the 20 graph Heawood family as the minor minimal graphs not 2--apex with at most 21 edges, providing new proofs for intrinsic knotting properties.

Findings

Heawood family graphs are exactly the minor minimal not 2--apex graphs with ≤21 edges.

Triangle-Y moves on K7 produce the minor minimal intrinsically knotted graphs.

Petersen family graphs are the minor minimal not apex graphs with ≤17 edges.

Abstract

We show that the 20 graph Heawood family, obtained by a combination of triangle-Y and Y-triangle moves on $K_7$, is precisely the set of graphs of at most 21 edges that are minor minimal for the property not $2$--apex. As a corollary, this gives a new proof that the 14 graphs obtained by triangle-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 for the property not apex.