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 P of n points admits 4-connected triangulation. We propose a necessary and sufficient condition for recognizing P , and present an O(n) 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) steps. This method provides a new insight to the structure of 4-connected triangulation of point sets.