Quadrangulations of a Polygon with Spirality

Abstract

Given an n-sided polygon P on the plane with \(n \ge 4\), a quadrangulation of P is a geometric plane graph such that the boundary of the outer face is P and that each finite face is quadrilateral. Clearly, P is quadrangulatable (i.e., admits a quadrangulation) only if n is even, but there is a non-quadrangulatable even-sided polygon. Ramaswami et al. [Comp Geom 9:257–276, (1998)] proved that every n-sided polygon P with \(n \ge 4\) even admits a quadrangulation with at most \(\lfloor \frac{n-2}{4} \rfloor\) Steiner points, where a Steiner point for P is an auxiliary point which can be put in any position in the interior of P. In this paper, introducing the notion of the spirality of P to control a structure of P (independent of n), we estimate the number of Steiner points to quadrangulate P.