Minimum Cuts in Geometric Intersection Graphs
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Let 𝒟 be a set of n disks in the plane. The disk graph G_𝒟 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 graph G^→_𝒟 for 𝒟 is the directed graph with vertex set 𝒟 in which there is an edge from a disk D_1 ∈𝒟 to a disk D_2 ∈𝒟 if and only if D_1 contains the center of D_2. Given 𝒟 and two non-intersecting disks s, t ∈𝒟, we show that a minimum s-t vertex cut in G_𝒟 or in G^→_𝒟 can be found in O(n^3/2polylog n) 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, L_ℓ and L_r, 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(n^3/2polylog n) expected time.
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