# The Pachner graph of 2-spheres

## TL;DR

This paper explores the connectivity properties of various subgraphs within the Pachner graph of 2-spheres, revealing both connected and disconnected substructures based on specific sphere classes.

## Contribution

It demonstrates the connectivity of the subgraph of flag 2-spheres and identifies multiple components in the subgraph of stacked 2-spheres, highlighting structural differences.

## Key findings

- Flag 2-sphere subgraph is connected.
- Stacked 2-sphere subgraph has multiple components.
- Connectivity varies with sphere class.

## Abstract

It is well-known that the Pachner graph of $n$-vertex triangulated $2$-spheres is connected, i.e., each pair of $n$-vertex triangulated $2$-spheres can be turned into each other by a sequence of edge flips for each $n\geq 4$. In this article, we study various induced subgraphs of this graph. In particular, we prove that the subgraph of $n$-vertex flag $2$-spheres distinct from the double cone is still connected. In contrast, we show that the subgraph of $n$-vertex stacked $2$-spheres has at least as many connected components as there are trees on $\lfloor\frac{n-5}{3}\rfloor$ nodes with maximum node-degree at most four.