On pseudo-disk hypergraphs.

Let F be a family of pseudo-disks in the plane, and P be a finite subset of F. Consider the hyper-graph H(P,F) whose vertices are the pseudo-disks in P and the edges are all subsets of P of the form {D∈P|D∩S≠∅}, where S is a pseudo-disk in F. We give an upper bound of O(nk3) for the number of edges in H(P,F) of cardinality at most k. This generalizes a result of Buzaglo et al. [4].As an application of our bound, we obtain an algorithm that computes a constant-factor approximation to the minimum-weight dominating set in a collection of pseudo-disks in the plane, in expected polynomial time.