  We give a deterministic algorithm for finding the minimum (weight) cut of an
undirected graph on $n$ vertices and $m$ edges using $\text{polylog}(n)$ calls
to any maximum flow subroutine. Using the current best deterministic maximum
flow algorithms, this yields an overall running time of $\tilde O(m \cdot
\min(\sqrt{m}, n^{2/3}))$ for weighted graphs, and $m^{4/3+o(1)}$ for
unweighted (multi)-graphs. This marks the first improvement for this problem
since a running time bound of $\tilde O(mn)$ was established by several papers
in the early 1990s.
  Our global minimum cut algorithm is obtained as a corollary of a minimum
Steiner cut algorithm, where a minimum Steiner cut is a minimum (weight) set of
edges whose removal disconnects at least one pair of vertices among a
designated set of terminal vertices. The running time of our deterministic
minimum Steiner cut algorithm matches that of the global minimum cut algorithm
stated above. Using randomization, the running time improves to $m^{1+o(1)}$
because of a faster maximum flow subroutine; this improves the best known
randomized algorithm for the minimum Steiner cut problem as well.
  Our main technical contribution is a new tool that we call *isolating cuts*.
Given a set of vertices $R$, this entails finding cuts of minimum weight that
separate (or isolate) each individual vertex $v\in R$ from the rest of the
vertices $R\setminus \{v\}$. Na\"ively, this can be done using $|R|$ maximum
flow calls, but we show that just $O(\log |R|)$ suffice for finding isolating
cuts for any set of vertices $R$. We call this the *isolating cut lemma*.
