# A Short Proof of the Toughness of Delaunay Triangulations

## TL;DR

This paper provides a concise, self-contained proof that Delaunay triangulations of planar point sets are 1-tough, implying they have perfect matchings and confirming a conjecture about blocking points.

## Contribution

It offers a new, shorter proof of the toughness of Delaunay triangulations and verifies a related conjecture on blocking points.

## Key findings

- Delaunay triangulations are 1-tough for planar point sets.
- They have guaranteed perfect matchings.
- At least n points are needed to block any n-vertex Delaunay triangulation.

## Abstract

We present a self-contained short proof of the seminal result of Dillencourt (SoCG 1987 and DCG 1990) that Delaunay triangulations, of planar point sets in general position, are 1-tough. An important implication of this result is that Delaunay triangulations have perfect matchings. Another implication of our result is a proof of the conjecture of Aichholzer et al. (2010) that at least $n$ points are required to block any $n$-vertex Delaunay triangulation