Vertex Separation in Networks

We study the problems of finding a subset of nodes having a given size $k$ and satisfying one of the following separation properties: The set is disconnected from the rest of the graph by a small/large cut or by a small separator. The considered problems are of interest in several practical settings, such as epidemiology or disaster control as well as to contrast viruses, malware, or misinformation propagate quickly in online social networks. All the considered problems are known to be NP-hard. Being computation time for very large networks is an important issue, we consider some parameters of the input graph $G$ and show that the problems become tractable for small values of such parameters. Namely, we show that they become tractable when parameterized either by the neighborhood diversity or by the treewidth of G. We also consider the complexity of the problems when parameterized by the clique-width cw of $G$ and show that they all can be solved in $O(n^{f(\text{cw})})$, where $n$ is the number of nodes in G. We also show that there is no $f(\text{cw})n^{o(\text{cw})-}$ time algorithm for the graph cut problems (unless ETH fails).