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Suppose $G$ is a bipartite graph which has a perfect matching. I want to find the fewest number of edges to delete from $G$ so that a perfect matching no longer exists. What is the complexity of this problem?

I'm aware that perfect matching has a formulation as a max-flow. One approach is, therefore, to find a min-cut in a corresponding graph, and see how many edges need to be removed so that the max-flow cannot carry as much flow. The problem with this approach is that a different min-cut might require the deletion of fewer edges, so it seems like this involves looking at all the min cuts in a graph.

Raphael
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1 Answers1

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This problem is known as the preclusion number of a graph, or minimum blocker perfect matching problem. It was shown to be $\mathcal{NP}$-hard in Lacroix, Mathieu; Mahjoub, A. Ridha; Martin, Sébastien; Picouleau, Christophe, On the NP-completeness of the perfect matching free subgraph problem, Theor. Comput. Sci. 423, 25-29 (2012). ZBL1237.68089.

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