Worst-case Deterministic Fully-Dynamic Planar 2-vertex Connectivity

TL;DR

This paper presents a deterministic data structure for fully-dynamic planar graphs that efficiently maintains 2-vertex connectivity information with worst-case $O( ext{log}^2 n)$ update time, improving upon previous amortized bounds.

Contribution

It introduces a new worst-case efficient data structure for maintaining 2-vertex connectivity in planar graphs under various updates, with deterministic guarantees.

Findings

Supports all updates and queries in $O( ext{log}^2 n)$ worst-case time.

Maintains a combinatorial embedding of the graph dynamically.

Achieves a significant improvement over previous amortized bounds for planar biconnectivity.

Abstract

We study dynamic planar graphs with $n$ vertices, subject to edge deletion, edge contraction, edge insertion across a face, and the splitting of a vertex in specified corners. We dynamically maintain a combinatorial embedding of such a planar graph, subject to connectivity and $2$-vertex-connectivity (biconnectivity) queries between pairs of vertices. Whenever a query pair is connected and not biconnected, we find the first and last cutvertex separating them. Additionally, we allow local changes to the embedding by flipping the embedding of a subgraph that is connected by at most two vertices to the rest of the graph. We support all queries and updates in deterministic, worst-case, $O(\log^2 n)$ time, using an $O(n)$-sized data structure. Previously, the best bound for fully-dynamic planar biconnectivity (subject to our set of operations) was an amortised $\tilde{O}(\log^3 n)$ for general graphs, and algorithms with worst-case polylogarithmic update times were known only in the partially dynamic (insertion-only or deletion-only) setting.