How to find the minimal cut in a network?

To find the minimal cut in a network, we can use the Max-Flow Min-Cut Theorem.

The Max-Flow Min-Cut Theorem states that the maximum flow in a network is equal to the minimum capacity of the cut. A cut is a partition of the vertices into two disjoint sets, such that the source vertex is in one set and the sink vertex is in the other set. The capacity of a cut is the sum of the capacities of the edges crossing the cut from the source set to the sink set.

To find the minimal cut, we can first find the maximum flow using a flow algorithm such as the Ford-Fulkerson algorithm or the Edmonds-Karp algorithm. Once we have found the maximum flow, we can use the residual graph to find the minimal cut.

The residual graph is a graph that shows the remaining capacity of each edge after the flow has been sent through the network. We can use a depth-first search or a breadth-first search to find the vertices that are reachable from the source vertex in the residual graph. These vertices form one set of the cut, and the remaining vertices form the other set of the cut.

The capacity of the cut is equal to the sum of the capacities of the edges from the source set to the sink set in the original graph. This is the minimal cut.

The capacity of the cut is equal to the sum of the capacities of the edges from the source set to the sink set in the original graph. This is the minimal cut. For more on how such principles apply in practical scenarios, see Real World Applications of Graph Analysis.

A-Level Maths Tutor Summary: To find the minimal cut in a network, use the Max-Flow Min-Cut Theorem. First, calculate the maximum flow with algorithms like Ford-Fulkerson or Edmonds-Karp. Then, identify the minimal cut by examining the residual graph, which reveals the remaining capacity after the flow. The cut splits the vertices into two groups, with the minimal cut's capacity being the smallest sum of crossing edges. To explore further the mathematical underpinnings of these techniques, consider our notes on Introduction to Integrals and Types of Numbers.