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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

Acknowledgements

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

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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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