Good $r$-divisions Imply Optimal Amortised Decremental Biconnectivity

TL;DR

This paper introduces a highly efficient data structure for decremental biconnectivity that preprocesses graphs quickly and supports constant-time connectivity queries, improving performance for large graph classes.

Contribution

It provides the first optimal amortized decremental biconnectivity data structure with constant query time for graphs with suitable r-divisions.

Findings

Supports edge-deletions in O(m) total time

Answers pairwise biconnectivity queries in O(1) worst-case time

Applies to planar and minor-free graphs

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.