A graph reduction step preserving element-connectivity and applications

Given an undirected graph G=(V,E) and subset of terminals T⊆V, the element-connectivity κ́ _G (u,v) of two terminals u,v T is the maximum number of u-v paths that are pairwise disjoint in both edges and non-terminals V\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 [18] 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 the step to connectivity and network design problems: First, we show a polylogarithmic approximation for the problem of packing element-disjoint Steiner forests in general graphs, and an O(1)-approximation in planar graphs. Second, we find a very short and intuitive proof of a spider-decomposition theorem of Chuzhoy and Khanna [10] in the context of the single-sink k-vertex-connectivity problem. Our results highlight the effectiveness of the element-connectivity reduction step; we believe it will find more applications in the future.