  We present an algorithm for distributed networks to efficiently find a small
vertex cut in the CONGEST model. Given a positive integer $\kappa$, our
algorithm can, with high probability, either find $\kappa$ vertices whose
removal disconnects the network or return that such $\kappa$ vertices do not
exist. Our algorithm takes $\kappa^3\cdot \tilde{O}(D+\sqrt{n})$ rounds, where
$n$ is the number of vertices in the network and $D$ denotes the network's
diameter. This implies $\tilde{O}(D+\sqrt{n})$ round complexity whenever
$\kappa=\text{polylog}(n)$.
  Prior to our result, a bound of $\tilde{O}(D)$ is known only when
$\kappa=1,2$ [Parter, Petruschka DISC'22]. For $\kappa\geq 3$, this bound can
be obtained only by an $O(\log n)$-approximation algorithm [Censor-Hillel,
Ghaffari, Kuhn PODC'14], and the only known exact algorithm takes
$O\left((\kappa\Delta D)^{O(\kappa)}\right)$ rounds, where $\Delta$ is the
maximum degree [Parter DISC'19]. Our result answers an open problem by
Nanongkai, Saranurak, and Yingchareonthawornchai [STOC'19].
