  Given a large edge-weighted network $G$ with $k$ terminal vertices, we wish
to compress it and store, using little memory, the value of the minimum cut (or
equivalently, maximum flow) between every bipartition of terminals. One
appealing methodology to implement a compression of $G$ is to construct a
\emph{mimicking network}: a small network $G'$ with the same $k$ terminals, in
which the minimum cut value between every bipartition of terminals is the same
as in $G$. This notion was introduced by Hagerup, Katajainen, Nishimura, and
Ragde [JCSS '98], who proved that such $G'$ of size at most $2^{2^k}$ always
exists. Obviously, by having access to the smaller network $G'$, certain
computations involving cuts can be carried out much more efficiently.
  We provide several new bounds, which together narrow the previously known gap
from doubly-exponential to only singly-exponential, both for planar and for
general graphs. Our first and main result is that every $k$-terminal planar
network admits a mimicking network $G'$ of size $O(k^2 2^{2k})$, which is
moreover a minor of $G$. On the other hand, some planar networks $G$ require
$|E(G')| \ge \Omega(k^2)$. For general networks, we show that certain bipartite
graphs only admit mimicking networks of size $|V(G')| \geq 2^{\Omega(k)}$, and
moreover, every data structure that stores the minimum cut value between all
bipartitions of the terminals must use $2^{\Omega(k)}$ machine words.
