Four-connected triangulations of planar point sets

TL;DR

This paper presents a polynomial-time algorithm to determine if a planar point set admits a 4-connected triangulation, providing a necessary and sufficient condition and advancing understanding in geometric graph theory.

Contribution

It introduces the first polynomial-time algorithm for recognizing 4-connected triangulations and offers new structural insights into such triangulations.

Findings

Polynomial-time recognition algorithm for 4-connected triangulations

Necessary and sufficient condition for 4-connected triangulation recognition

A simple method for constructing noncomplex triangulations

Abstract

In this paper, we consider the problem of determining in polynomial time whether a given planar point set $P$ of $n$ points admits 4-connected triangulation. We propose a necessary and sufficient condition for recognizing $P$, and present an $O(n^3)$ algorithm of constructing a 4-connected triangulation of $P$. Thus, our algorithm solves a longstanding open problem in computational geometry and geometric graph theory. We also provide a simple method for constructing a noncomplex triangulation of $P$ which requires $O(n^2)$ steps. This method provides a new insight to the structure of 4-connected triangulation of point sets.