The edge connectivity of an undirected graph is the minimum number k of edgesthat must be removed to disconnect the graph. For example, the edge connectivityof a tree is 1, and the edge connectivity of a cyclic chain of vertices is 2. Showhow to determine the edge connectivity of an undirected graph G D .V; E/ byrunning a maximum-flow algorithm on at most jV j flow networks, each havingO.V / vertices and O.E/ edges.
Question
The edge connectivity of an undirected graph is the minimum number k of edgesthat must be removed to disconnect the graph. For example, the edge connectivityof a tree is 1, and the edge connectivity of a cyclic chain of vertices is 2. Showhow to determine the edge connectivity of an undirected graph G D .V; E/ byrunning a maximum-flow algorithm on at most jV j flow networks, each havingO.V / vertices and O.E/ edges.
Solution
To determine the edge connectivity of an undirected graph G(V, E) using a maximum-flow algorithm, follow these steps:
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Choose an arbitrary vertex v from the graph G.
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Construct a flow network for each of the remaining vertices u in the graph. In each flow network, v and u are the source and sink, respectively.
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For each edge (x, y) in the original graph, create two directed edges (x, y) and (y, x) in the flow network. Assign each edge in the flow network a capacity of 1.
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Run a maximum-flow algorithm (like Ford-Fulkerson or Edmonds-Karp) on each of the flow networks. The maximum flow of a network is the maximum amount of flow that can be pushed from the source to the sink without violating the capacity constraints.
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The edge connectivity of the original graph G is the minimum maximum flow among all the flow networks. This is because the edge connectivity is defined as the minimum number of edges that need to be removed to disconnect the graph, which corresponds to the minimum maximum flow.
This process runs a maximum-flow algorithm on at most |V| flow networks, each having O(V) vertices and O(E) edges, as required.
Similar Questions
True or False: If any edge is removed from a tree, then the resulting graph is not connected.
Alice and Bob have an undirected graph of n nodes and three types of edges:Type 1: Can be traversed by Alice only.Type 2: Can be traversed by Bob only.Type 3: Can be traversed by both Alice and Bob.Given an array edges where edges[i] = [typei, ui, vi] represents a bidirectional edge of type typei between nodes ui and vi, find the maximum number of edges you can remove so that after removing the edges, the graph can still be fully traversed by both Alice and Bob. The graph is fully traversed by Alice and Bob if starting from any node, they can reach all other nodes.Return the maximum number of edges you can remove, or return -1 if Alice and Bob cannot fully traverse the graph. Example 1:Input: n = 4, edges = [[3,1,2],[3,2,3],[1,1,3],[1,2,4],[1,1,2],[2,3,4]]Output: 2Explanation: If we remove the 2 edges [1,1,2] and [1,1,3]. The graph will still be fully traversable by Alice and Bob. Removing any additional edge will not make it so. So the maximum number of edges we can remove is 2.Example 2:Input: n = 4, edges = [[3,1,2],[3,2,3],[1,1,4],[2,1,4]]Output: 0Explanation: Notice that removing any edge will not make the graph fully traversable by Alice and Bob.Example 3:Input: n = 4, edges = [[3,2,3],[1,1,2],[2,3,4]]Output: -1Explanation: In the current graph, Alice cannot reach node 4 from the other nodes. Likewise, Bob cannot reach 1. Therefore it's impossible to make the graph fully traversable. Constraints:1 <= n <= 1051 <= edges.length <= min(105, 3 * n * (n - 1) / 2)edges[i].length == 31 <= typei <= 31 <= ui < vi <= nAll tuples (typei, ui, vi) are distinct.
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