Four-Connected Triangulations of Planar Point Sets

Abstract

In this paper, we consider the problem of determining in polynomial time whether a given planar point set $$P$$P of $$n$$n points in general position admits a 4-connected triangulation. We propose a necessary and sufficient condition for recognizing such point sets $$P$$P, and present an $$O(n^3)$$O(n3) time algorithm for constructing a 4-connected triangulation of $$P$$P, if it exists. Thus, our algorithm solves a longstanding open problem in computational geometry and geometric graph theory. We also provide a simple $$O(n^2)$$O(n2) time method for constructing a non-complex triangulation of $$P$$P, if it exists. This method provides a new insight into the structure of 4-connected triangulations of point sets.