  We study how to vertex-sparsify a graph while preserving both the graph's
metric and structure. Specifically, we study the Steiner point removal (SPR)
problem where we are given a weighted graph $G=(V,E,w)$ and terminal set $V'
\subseteq V$ and must compute a weighted minor $G'=(V',E', w')$ of $G$ which
approximates $G$'s metric on $V'$. A major open question in the area of metric
embeddings is the existence of $O(1)$ multiplicative distortion SPR solutions
for every (non-trivial) minor-closed family of graphs. To this end prior work
has studied SPR on trees, cactus and outerplanar graphs and showed that in
these graphs such a minor exists with $O(1)$ distortion.
  We give $O(1)$ distortion SPR solutions for series-parallel graphs, extending
the frontier of this line of work. The main engine of our approach is a new
metric decomposition for series-parallel graphs which we call a hammock
decomposition. Roughly, a hammock decomposition is a forest-like structure that
preserves certain critical parts of the metric induced by a series-parallel
graph.
