Hi everyone!

There is a somewhat well-known result about bipartite graphs which is formulated as follows:

In any bipartite graph, the number of edges in a maximum matching equals the number of vertices in a minimum vertex cover.

Knowing it you can easily find the size of a minimum vertex cover in a given bipartite graph. But what about actually recovering such cover? Well... There is an algorithm to it, but it's lengthy and not very motivated, so I constantly forget its details. However, there is a simple alternative, which is easily reproduced from scratch, even if you forget specific details.

It is most likely well known to those familiar with the topic, but still might be of interest to someone.

Recall that given a flow network $$$G=(V, E)$$$, finding minimum cut between $$$s$$$ and $$$t$$$ is done as follows:

• Find a maximum flow $$$f$$$ from $$$s$$$ to $$$t$$$ in the network;
• Find $$$X$$$, the set of all vertices reachable from $$$s$$$ in the residual network;
• Minimum cut is formed by $$$S=X$$$ and $$$T=V \setminus X$$$, that is the minimum cut is formed by the edges going from $$$S$$$ to $$$T$$$.

Now, recall that bipartite matching between sets of vertices $$$A$$$ and $$$B$$$ may be found as maximum flow in the following network:

• There is an edge of capacity $$$1$$$ going from $$$s$$$ to every vertex of $$$A$$$;
• There is an edge of capacity $$$1$$$ going from every vertex of $$$B$$$ to $$$t$$$;
• There is an edge of capacity $$$+\infty$$$ going between some vertices of $$$A$$$ and $$$B$$$, as defined by the bipartite graph.

Now... Is there anything special about minimum cut in such network?

Minimum cut $$$(S, T)$$$ in a flow network induced by a bipartite graph

Generally, some vertices from $$$A$$$ and some vertices from $$$B$$$ will be with the source $$$s$$$ in the cut, while other will be with the sink $$$t$$$.

Let $$$A = A_S \cup A_T$$$ and $$$B = B_S \cup B_T$$$, such that $$$A_S, B_S \subset S$$$ and $$$A_T, B_T \subset T$$$. Then the following is evidently true:

• There are no edges from $$$A_S$$$ to $$$B_T$$$ (otherwise the cut would be infinite);
• Thus, every edge in the bipartite graph is incident to some vertex from $$$A_T$$$ or $$$B_S$$$;
• Minimum cut is formed only by edges going from $$$S$$$ to $$$A_T$$$ and from $$$B_S$$$ to $$$T$$$ and thus its size is $$$|A_T|+|B_S|$$$.

Putting it all together, the minimum vertex cover is $$$A_T \cup B_S$$$, and it can be easily found from the minimum cut:

1. Find a minimum cut $$$(S, T)$$$ in the flow network of the maximum matching on bipartite graph with parts $$$A$$$ and $$$B$$$;
2. A minimum vertex cover is comprised of the sets $$$A_T = (A \cap T)$$$ and $$$B_S = (B \cap S)$$$.