Flips in Two-dimensional Hypertriangulations

TL;DR

This paper investigates the structure of hypertriangulations in planar point sets, introducing new flip operations and proving connectivity for level-2 hypertriangulations, advancing understanding of their combinatorial properties.

Contribution

It introduces four types of flips in hypertriangulations and proves the connectivity of level-2 hypertriangulations via these flips.

Findings

Four types of flips in hypertriangulations are defined.

Level-2 hypertriangulations are connected through these flips.

Provides a new framework for understanding hypertriangulation transformations.

Abstract

We study flips in hypertriangulations of planar points sets. Here a level-$k$ hypertriangulation of $n$ points in the planes is a subdivision induced by the projection of a $k$-hypersimplex, which is the convex hull of the barycenters of the $(k-1)$-dimensional faces of the standard $(n-1)$-simplex. In particular, we introduce four types of flips and prove that the level-2 hypertriangulations are connected by these flips.