On the parameterized complexity of cutting a few vertices from a graph

We study the parameterized complexity of separating a small set of vertices from a graph by a small vertex-separator. That is, given a graph $G$ and integers $k$, $t$, the task is to find a vertex set $X$ with $|X| \le k$ and $|N(X)| \le t$. We show that - the problem is fixed-parameter tractable (FPT) when parameterized by $t$ but W[1]-hard when parameterized by $k$, and - a terminal variant of the problem, where $X$ must contain a given vertex $s$, is W[1]-hard when parameterized either by $k$ or by $t$ alone, but is FPT when parameterized by $k + t$. We also show that if we consider edge cuts instead of vertex cuts, the terminal variant is NP-hard.