On disconnected cuts and separators

Abstract

For a connected graph $G=(V,E)$, a subset $U\subseteq V$ is called a disconnected cut if $U$ disconnects the graph, and the subgraph induced by $U$ is disconnected as well. A natural condition is to impose that for any $u\in U$, the subgraph induced by $(V\setminus U)\cup \left\{u\right\}$ is connected. In that case, $U$ 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, $s$ and $t$, is NP-complete.