Partitions of planar point sets into polygons

TL;DR

This paper characterizes when planar point sets can be partitioned into polygons of specified sizes, provides a polynomial-time algorithm for such partitions, and explores the computational complexity of related problems.

Contribution

It introduces a polynomial-time algorithm for partitioning point sets into polygons of given sizes and characterizes the problem via visibility graph factors.

Findings

Partitioning into specified polygons is characterized by visibility graph factors.

Polynomial-time algorithm exists for certain partitioning problems.

Finding K_k-factors for k ≥ 5 is NP-hard.

Abstract

In this paper, we characterize planar point sets that can be partitioned into disjoint polygons of arbitrarily specified sizes. We provide an algorithm to construct such a partition, if it exists, in polynomial time. We show that this problem is equivalent to finding a specified $2$-factor in the visibility graph of the point set. The characterization for the case where all cycles have length $3$ also translates to finding a $K_3$-factor of the visibility graph of the point set. We show that the generalized problem of finding a $K_k$-factor of the visibility graph of a given point set for $k \geq 5$ is NP-hard.