Graphs on 21 edges that are not 2--apex

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.

Submitted to Involve

References in corpus (2)

• Intrinsic knotting and linking of almost complete graphs
• Intrinsically knotted graphs with 21 edges

Cited by in corpus (5)

• Six variations on a theme: almost planar graphs
• The $K_{n+5}$ and $K_{3^2,1^n}$ families are obstructions to $n$-apex
• More intrinsically knotted graphs with 22 edges and the restoring method
• On intrinsically knotted and linked graphs
• The minor minimal intrinsically chiral graphs