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

Abstract

Abstract. 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 complexity. Our upper bounds depend on new Ramsey-type results concerning disjoint empty convex k-gons in point sets.

Disjoint empty convex pentagons in planar point sets

“…The results H(3, 4) = 7 and H(4, 5) ≤ 14 were later reconfirmed by Wu and Ding [33]. Using the computer-aided order-type enumeration method, Aichholzer et al [1] proved that every set of 11 points in the plane, no three on a line, contains either a 6-hole or a 5-hole and a disjoint 4-hole. Recently, this result was proved geometrically by Bhattacharya and Das [5,6].…”

“…Later, Xu and Ding [34] improved the lower bound to ⌈ n+1 4 ⌉. Recently, Aichholzer et al [1] introduced the notion pseudo-convex partitioning of planar point sets, which extends the concept partitioning, in the sense, that they allow both convex polygons and pseudo-triangles in the partition.…”

On pseudo-convex partitions of a planar point set

“…Aichholzer et al [1] showed that ψ(n) ≤ ⌈ n 4 ⌉. They also observed that 3 ≤ ψ(13) ≤ 4, and mention the possibility of a better upper bound of ⌈ 3n 13 ⌉ on ψ(n) by conjecturing that ψ(13) = 3.…”

“…Any two empty pseudo-triangles, or a hole and an empty pseudotriangle are said to be disjoint if their vertex sets as well as their interiors are disjoint. Recently, Aichholzer et al [1] introduced the problem of partitioning planar point sets with disjoint holes or empty pseudo-triangles. Given a set S of n points in the plane, no three on a line, a pseudo-convex partition of S is a partition of S into subsets S 1 , S 2 , .…”

“…The upper bound follows from the simple observation that every set of 4 points forms either an 4-hole or an empty 4-pseudo-triangle. In fact, using computer-aided search, Aichholzer et al [1] obtained bounds on the pseudo-convex partition number ψ(n) for small point sets. However, they were unable to find the exact value of ψ (13), and mentioned the possibility of a non-trivial upper bound on ψ(n) by conjecturing that ψ(13) = 3.…”