Counting Convex Polygons in Planar Point Sets

On Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-Triangles

We propose a novel subdivision of the plane that consists of both convex polygons and pseudo-triangles. This pseudo-convex decomposition is significantly sparser than either convex decompositions or pseudo-triangulations for planar point sets and simple polygons. We also introduce pseudo-convex partitions and coverings. We establish some basic properties and give combinatorial bounds on their c...

Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-triangles

We propose a novel subdivision of the plane that consists of both convex polygons and pseudo-triangles. This pseudo-convex decomposition is significantly sparser than either convex decompositions or pseudo-triangulations for planar point sets and simple polygons. We also introduce pseudo-convex partitions and coverings. We establish some basic properties and give combinatorial bounds on their c...

Planar Point Sets With Large Minimum Convex Decompositions

We show the existence of sets with n points (n > 4) for which every convex decomposition contains more than f§« — § polygons, which refutes the conjecture that for every set of n points there is a convex decomposition with at most n + C polygons. For sets having exactly three extreme points we show that more than n + s/2(n 3) 4 polygons may be necessary to form a convex decomposition.

Counting Convex Polygons in Planar

On pseudo-convex decompositions, partitions, and coverings

We introduce pseudo-convex decompositions, partitions, and coverings for planar point sets. They are natural extensions of their convex counterparts and use both convex polygons and pseudo-triangles. We discuss some of their basic combinatorial properties and establish upper and lower bounds on their complexity.