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Here I am trying to disconnect two vertices in a graph with minimum edge removal possible.

example graph In this graph between two vertices A and Z you can find the answer in many ways. In optimal way you can remove just one edge from A to B.

If there is any specific algorithm for this?

I found some suggestions to solve this by using maximum flow min cut problem but I am not getting general idea to convert this problem into max flow min cut theorem. Also in the process I might end up removing an edge between F and G which is useless.

try-catch-finally
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full_stuck_developer
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  • As you state it, this is exactly min-cut: Remove the minimum number of edges to fragment the graph.Since min-cut is dual to max-flow, any algorithm for solving one also solves the other. – dhke May 08 '16 at 08:21

1 Answers1

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This can be solved using Max Flow - Min Cut problem.

You can model your graph into network flow as follows:
1. Consider A to be the source vertex, Z to be the sink vertex.
2. Set the capacity of each edge to be 1 unit.

Now, solve the Max Flow - Min Cut problem in above network. With it you will be able to find number of edge-disjoint paths from A to Z. For every such path, remove the first edge (edge originating from source A).

Proof:
Observe that after removing above edges, you won't be able to reach A to Z. If you had a path, then Max flow algorithm would have included this path in set of edge-disjoint-paths.
Also, by construction of network, you can't remove lesser number of edges to disconnect A from Z

Rishit Sanmukhani
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