Contraction-Based Sparsification in Near-Linear Time
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Recently, Kawarabayashi and Thorup presented the first deterministic edge-connectivity recognition algorithm in near-linear time. A crucial step in their algorithm uses the existence of vertex subsets of a simple graph G on n vertices whose contractions leave a multigraph with Õ(n/δ) vertices and Õ(n) edges that preserves all non-trivial min-cuts of G. We show a very simple argument that improves this contraction-based sparsifier by eliminating the poly-logarithmic factors, that is, we show a contraction-based sparsification that leaves O(n/δ) vertices and O(n) edges, preserves all non-trivial min-cuts and can be computed in near-linear time Õ(|E(G)|). As consequence, every simple graph has O((n/δ)^2) non-trivial min-cuts. Our approach allows to represent all non-trivial min-cuts of a graph by a cactus representation, whose cactus graph has O(n/δ) vertices. Moreover, this cactus representation can be derived directly from the standard cactus representation of all min-cuts in linear time.
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