  Given an undirected graph G=(V,E) and subset of terminals T \subseteq V, the
element-connectivity of two terminals u,v \in T is the maximum number of u-v
paths that are pairwise disjoint in both edges and non-terminals V \setminus T
(the paths need not be disjoint in terminals). Element-connectivity is more
general than edge-connectivity and less general than vertex-connectivity. Hind
and Oellermann gave a graph reduction step that preserves the global
element-connectivity of the graph. We show that this step also preserves local
connectivity, that is, all the pairwise element-connectivities of the
terminals. We give two applications of this reduction step to connectivity and
network design problems:
  1. Given a graph G and disjoint terminal sets T_1, T_2, ..., T_m, we seek a
maximum number of element-disjoint Steiner forests where each forest connects
each T_i. We prove that if each T_i is k-element-connected then there exist
\Omega(\frac{k}{\log h \log m}) element-disjoint Steiner forests, where h =
|\bigcup_i T_i|. If G is planar (or more generally, has fixed genus), we show
that there exist \Omega(k) Steiner forests. Our proofs are constructive, giving
poly-time algorithms to find these forests; these are the first non-trivial
algorithms for packing element-disjoint Steiner Forests.
  2. We give a very short and intuitive proof of a spider-decomposition theorem
of Chuzhoy and Khanna in the context of the single-sink k-vertex-connectivity
problem; this yields a simple and alternative analysis of an O(k \log n)
approximation.
  Our results highlight the effectiveness of the element-connectivity reduction
step; we believe it will find more applications in the future.
