  In vertex-cut sparsification, given a graph $G=(V,E)$ with a terminal set
$T\subseteq V$, we wish to construct a graph $G'=(V',E')$ with $T\subseteq V'$,
such that for every two sets of terminals $A,B\subseteq T$, the size of a
minimum $(A,B)$-vertex-cut in $G'$ is the same as in $G$. In the most basic
setting, $G$ is unweighted and undirected, and we wish to bound the size of
$G'$ by a function of $k=|T|$. Kratsch and Wahlstr\"om [JACM 2020] proved that
every graph $G$ (possibly directed), admits a vertex-cut sparsifier $G'$ with
$O(k^3)$ vertices, which can in fact be constructed in randomized polynomial
time.
  We study (possibly directed) graphs $G$ that are quasi-bipartite, i.e., every
edge has at least one endpoint in $T$, and prove that they admit a vertex-cut
sparsifier with $O(k^2)$ edges and vertices, which can in fact be constructed
in deterministic polynomial time. In fact, this bound naturally extends to all
graphs with a small separator into bounded-size sets. Finally, we prove
information-theoretically a nearly-matching lower bound, i.e., that
$\tilde{\Omega}(k^2)$ edges are required to sparsify quasi-bipartite undirected
graphs.
