# Branch and cut algorithms for detecting critical nodes in undirected graphs

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## Abstract

In this paper we deal with the critical node problem, where a given number of nodes has to be removed from an undirected graph in order to maximize the disconnections between the node pairs of the graph. We propose an integer linear programming model with a non-polynomial number of constraints but whose linear relaxation can be solved in polynomial time. We derive different valid inequalities and some theoretical results about them. We also propose an alternative model based on a quadratic reformulation of the problem. Finally, we perform many computational experiments and analyze the corresponding results.

### Keywords

Critical node problem Branch and cut Valid inequalities Reformulation-linearization technique## Notes

### Acknowledgements

The authors would like to thank two anonymous referees, whose constructive comments helped to improve the initial version of the paper.

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