# Counting Polygon Triangulations is Hard

## TL;DR

This paper proves that counting the number of triangulations of a polygon is computationally very hard, specifically  extbackslash mathsf ext{ extbackslash P}-complete, even for polygons that are not simple.

## Contribution

It establishes the  extbackslash mathsf ext{ extbackslash P}-completeness of counting polygon triangulations, extending the understanding of computational complexity in geometric enumeration problems.

## Key findings

- Counting polygon triangulations is  extbackslash mathsf ext{ extbackslash P}-complete.
- The result applies to non-simple polygons.
- This advances complexity theory in computational geometry.

## Abstract

We prove that it is $\#\mathsf{P}$-complete to count the triangulations of a (non-simple) polygon.