Four-Connected Triangulations of Planar Point Sets

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