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• W2950308273 abstract "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." @default.
• W2950308273 created "2019-06-27" @default.
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• W2950308273 date "2009-02-16" @default.
• W2950308273 modified "2023-09-27" @default.
• W2950308273 title "A Graph Reduction Step Preserving Element-Connectivity and Applications" @default.
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