The Pachner graph of 2-spheres

Benjamin A. Burton, Basudeb Datta and Jonathan Spreer

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 $n\ge 4$. 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.

AMS Subject Classification: Primary 57Q15; secondary 57M20, 05C10.