Maximal k-Edge-Connected Subgraphs in Almost-Linear Time for Small k

Authors Thatchaphol Saranurak , Wuwei Yuan

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Author Details

Thatchaphol Saranurak
  • University of Michigan, Ann Arbor, MI, USA
Wuwei Yuan
  • Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China

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Thatchaphol Saranurak and Wuwei Yuan. Maximal k-Edge-Connected Subgraphs in Almost-Linear Time for Small k. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 92:1-92:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We give the first almost-linear time algorithm for computing the maximal k-edge-connected subgraphs of an undirected unweighted graph for any constant k. More specifically, given an n-vertex m-edge graph G = (V,E) and a number k = log^o(1) n, we can deterministically compute in O(m+n^{1+o(1)}) time the unique vertex partition {V_1,… ,V_z} such that, for every i, V_i induces a k-edge-connected subgraph while every superset V'_i ⊃ V_{i} does not. Previous algorithms with linear time work only when k ≤ 2 [Tarjan SICOMP'72], otherwise they all require Ω(m+n√n) time even when k = 3 [Chechik et al. SODA'17; Forster et al. SODA'20]. Our algorithm also extends to the decremental graph setting; we can deterministically maintain the maximal k-edge-connected subgraphs of a graph undergoing edge deletions in m^{1+o(1)} total update time. Our key idea is a reduction to the dynamic algorithm supporting pairwise k-edge-connectivity queries [Jin and Sun FOCS'20].

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Graph algorithms
  • Maximal k-edge-connected subgraphs
  • Dynamic k-edge connectivity


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