Minimum cuts in geometric intersection graphs

Abstract

Let D be a set of n disks in the plane. The disk graphGD for D is the undirected graph with vertex set D in which two disks are joined by an edge if and only if they intersect. The directed transmission graphGD→ for D is the directed graph with vertex set D in which there is an edge from a disk D1∈D to a disk D2∈D if and only if D1 contains the center of D2.Given D and two non-intersecting disks s,t∈D, we show that a minimum s-t vertex cut in GD or in GD→ can be found in O(n3/2polylogn) expected time. To obtain our result, we combine an algorithm for the maximum flow problem in general graphs with dynamic geometric data structures to manipulate the disks.As an application, we consider the barrier resilience problem in a rectangular domain. In this problem, we have a vertical strip S bounded by two vertical lines, Lℓ and Lr, and a collection D of disks. Let a be a point in S above all disks of D, and let b a point in S below all disks of D. The task is to find a curve from a to b that lies in S and that intersects as few disks of D as possible. Using our improved algorithm for minimum cuts in disk graphs, we can solve the barrier resilience problem in O(n3/2polylogn) expected time.