For a connected graph , a subset is called a disconnected cut if disconnects the graph, and the subgraph induced by is disconnected as well. A natural condition is to impose that for any , the subgraph induced by is connected. In that case, is called a minimal disconnected cut. We show that the problem of testing whether a graph has a minimal disconnected cut is NP-complete. We also show that the problem of testing whether a graph has a disconnected cut separating two specified vertices, and , is NP-complete.
Some of the results in this paper appeared in an extended abstract [6] presented at the 20th International Symposium on Algorithms and Computation (ISAAC 2009).