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