Dips at small sizes for topological graph obstruction sets

TL;DR

This paper investigates the minimal obstructions for topological graph properties, revealing a surprising dip in the number of obstructions at small sizes and providing classifications for specific cases.

Contribution

It identifies a significant reduction in the number of obstructions at size 23 for knotless embedding and classifies all obstructions for order ten graphs.

Findings

Only three obstructions to knotless embedding at size 23.

Fewer obstructions at size 23 than at size 22 or larger sizes.

Complete classification of obstructions for order ten graphs.

Abstract

The Graph Minor Theorem of Robertson and Seymour implies a finite set of obstructions for any minor closed graph property. We show that there are only three obstructions to knotless embedding of size 23, which is far fewer than the 92 of size 22 and the hundreds known to exist at larger sizes. We describe several other topological properties whose obstruction set demonstrates a similar dip at small size. For order ten graphs, we classify the 35 obstructions to knotless embedding and the 49 maximal knotless graphs.