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Good r-Divisions Imply Optimal Amortized Decremental Biconnectivity

Authors Jacob Holm , Eva Rotenberg

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LIPIcs.STACS.2021.42.pdf
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ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Dynamic graphs
  • 2-connectivity
  • graph minors
  • r-divisions
  • graph separators

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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.

Cite As Get 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

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.

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