Quadrangulations of a Polygon with Spirality

Given an n -sided polygon P on the plane with n ≥ 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 ≥ 4 even admits a quadrangulation with at most ⌊n-2/4⌋ 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 .