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Good r-Divisions Imply Optimal Amortized Decremental Biconnectivity

Authors Jacob Holm , Eva Rotenberg

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Author Details

Jacob Holm
  • University of Copenhagen, Denmark
Eva Rotenberg
  • Technical University of Denmark, Lyngby, Denmark

Funding

  • Holm, Jacob: Partially supported by the VILLUM Foundation grant 16582, "BARC".
  • Rotenberg, Eva: Partially supported by Independent Research Fund Denmark grants 2020-2023 (9131-00044B) "Dynamic Network Analysis" and 2018-2021 (8021-00249B), "AlgoGraph", and the VILLUM Foundation grant 37507 "Efficient Recomputations for Changeful Problems".

Acknowledgements

We are thankful to Adam Karczmarz and Jakub Łącki for their encouragement and interest in this work.

Cite AsGet BibTex

Jacob Holm and Eva Rotenberg. Good r-Divisions Imply Optimal Amortized Decremental Biconnectivity. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 42:1-42:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.42

Abstract

We present a data structure that, given a graph G of n vertices and m edges, and a suitable pair of nested r-divisions of G, preprocesses G in O(m+n) time and handles any series of edge-deletions in O(m) total time while answering queries to pairwise biconnectivity in worst-case O(1) time. In case the vertices are not biconnected, the data structure can return a cutvertex separating them in worst-case O(1) time. As an immediate consequence, this gives optimal amortized decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Dynamic graphs
  • 2-connectivity
  • graph minors
  • r-divisions
  • graph separators

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