  Given a graph where vertices are partitioned into $k$ terminals and
non-terminals, the goal is to compress the graph (i.e., reduce the number of
non-terminals) using minor operations while preserving terminal distances
approximately.The distortion of a compressed graph is the maximum
multiplicative blow-up of distances between all pairs of terminals. We study
the trade-off between the number of non-terminals and the distortion. This
problem generalizes the Steiner Point Removal (SPR) problem, in which all
non-terminals must be removed.
  We introduce a novel black-box reduction to convert any lower bound on
distortion for the SPR problem into a super-linear lower bound on the number of
non-terminals, with the same distortion, for our problem. This allows us to
show that there exist graphs such that every minor with distortion less than
$2~/~2.5~/~3$ must have $\Omega(k^2)~/~\Omega(k^{5/4})~/~\Omega(k^{6/5})$
non-terminals, plus more trade-offs in between. The black-box reduction has an
interesting consequence: if the tight lower bound on distortion for the SPR
problem is super-constant, then allowing any $O(k)$ non-terminals will not help
improving the lower bound to a constant.
  We also build on the existing results on spanners, distance oracles and
connected 0-extensions to show a number of upper bounds for general graphs,
planar graphs, graphs that exclude a fixed minor and bounded treewidth graphs.
Among others, we show that any graph admits a minor with $O(\log k)$ distortion
and $O(k^{2})$ non-terminals, and any planar graph admits a minor with
$1+\varepsilon$ distortion and $\widetilde{O}((k/\varepsilon)^{2})$
non-terminals.
