Minimum Weight Convex Quadrangulation of a Constrained Point Set

A convex quadrangulation with respect to a point set S is a planar subdivision whose vertices are the points of S, where the boundary of the unbounded outer face is the boundary of the convex hull of S, and every bounded interior face is convex and has four points from S on its boundary. A minimum weight convex quadrangulation with respect to S is a convex quadrangulation of S such that the sum of the Euclidean lengths of the edges of the subdivision is minimised. In this paper, we will present a polynomial time algorithm to determine whether a set of points S admits a convex quadrangulation if S is constrained to lie on a xed number of nested convex polygons, where the time complexity is polynomial in the cardinality of S. This algorithm can also be used to nd a minimum weight convex quadrangulation of the point set.