  We study the following version of cut sparsification. Given a large
edge-weighted network $G$ with $k$ terminal vertices, compress it into a
smaller network $H$ with the same terminals, such that every minimum terminal
cut in $H$ approximates the corresponding one in $G$, up to a factor $q\geq 1$
that is called the quality. (The case $q=1$ is known also as a mimicking
network). We provide new insights about the structure of minimum terminal cuts,
leading to new results for cut sparsifiers of planar graphs. Our first
contribution identifies a subset of the minimum terminal cuts, which we call
elementary, that generates all the others. Consequently, $H$ is a cut
sparsifier if and only if it preserves all the elementary terminal cuts (up to
this factor $q$). This structural characterization lead to improved bounds on
the size of $H$. For example, it improve the bound of mimicking-network size
for planar graphs into a near-optimal one. Our second and main contribution is
to refine the known bounds in terms of $\gamma=\gamma(G)$, which is defined as
the minimum number of faces that are incident to all the terminals in a planar
graph $G$. We prove that the number of elementary terminal cuts is
$O((2k/\gamma)^{2\gamma})$ (compared to $O(2^k)$ terminal cuts), and
furthermore obtain a mimicking-network of size $O(\gamma 2^{2\gamma} k^4)$,
which is near-optimal as a function of $\gamma$. In the analysis we break the
elementary terminal cuts into fragments, and count them carefully. Our third
contribution is a duality between cut sparsification and distance
sparsification for certain planar graphs, when the sparsifier $H$ is required
to be a minor of $G$. This duality connects problems that were previously
studied separately, implying new results, new proofs of known results, and
equivalences between open gaps.
