How to Secure Matchings Against Edge Failures
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The matching preclusion number of a graph is the minimal number of edges whose removal destroys all perfect matchings. We provide algorithms and hardness results for the task of increasing the matching preclusion number from one to two in bipartite graphs at minimal cost. Our motivation is to make matchings of a graph robust against the failure of a single edge. Our methods rely on a close relationship to the classical strong connectivity augmentation problem. For the unit weight problem we provide a deterministic _2 n-factor approximation algorithm, as well as polynomial-time algorithms for graphs of bounded treewidth and chordal-bipartite graphs. For general weights we prove a dichotomy theorem characterizing minor-closed graph classes which allow for a polynomial-time algorithm.
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