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arXiv · 2209.14079

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

Jacob Holm·Ivor van der Hoog·Eva Rotenberg
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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.

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Jacob Holm, Ivor van der Hoog, Eva Rotenberg. 2022-09-28. Worst-case Deterministic Fully-Dynamic Planar 2-vertex Connectivity. https://arxiv.org/abs/2209.14079

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