Blocking the $$k$$k-Holes of Point Sets in the Plane

Let $$P$$P be a set of $$n$$n points in the plane in general position. A subset $$H$$H of $$P$$P consisting of $$k$$k elements that are the vertices of a convex polygon is called a $$k$$k-hole of $$P$$P, if there is no element of $$P$$P in the interior of its convex hull. A set $$B$$B of points in the plane blocks the $$k$$k-holes of $$P$$P if any $$k$$k-hole of $$P$$P contains at least one element of $$B$$B in the interior of its convex hull. In this paper we establish upper and lower bounds on the sizes of $$k$$k-hole blocking sets, with emphasis in the case $$k=5$$k=5.