On the minimum number of mutually disjoint holes in planar point sets

Abstract

Let P be a set of n points in general position in the plane. In 1996, Urabe considered a partition of P into subsets S 1 ∪ · · · ∪ S l such that each S i forms a hole (or an empty convex polygon) of P and these holes are mutually disjoint. Let f (P ) be the minimum number of holes over all such partitions of P and F (n) = max{ f (P )} over all sets P of n points. Then the current best bounds are given by n+1 4 ≤ F (n) ≤ 5n 18 . In this paper, we prove that F (n) ≤ n 4 + 1.