Minimum Cuts in Geometric Intersection Graphs

05/02/2020
by   Sergio Cabello, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
07/03/2019

Triangles and Girth in Disk Graphs and Transmission Graphs

Let S ⊂R^2 be a set of n sites, where each s ∈ S has an associated radiu...
research
02/28/2023

On the geometric thickness of 2-degenerate graphs

A graph is 2-degenerate if every subgraph contains a vertex of degree at...
research
12/20/2017

Recognizing Generalized Transmission Graphs of Line Segments and Circular Sectors

Suppose we have an arrangement A of n geometric objects x_1, ..., x_n ⊆R...
research
06/27/2023

Insertion-Only Dynamic Connectivity in General Disk Graphs

Let S ⊆ℝ^2 be a set of n sites in the plane, so that every site s ∈ S ha...
research
12/09/2022

Breaking the Barrier 2^k for Subset Feedback Vertex Set in Chordal Graphs

The Subset Feedback Vertex Set problem (SFVS), to delete k vertices from...
research
10/09/2019

Hardness of Minimum Barrier Shrinkage and Minimum Activation Path

In the Minimum Activation Path problem, we are given a graph G with edge...
research
05/06/2019

Geometric Firefighting in the Half-plane

In 2006, Alberto Bressan suggested the following problem. Suppose a circ...

Please sign up or login with your details

Forgot password? Click here to reset