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(log² n) time, using an O(n)-sized data structure.
@InProceedings{holm_et_al:LIPIcs.SoCG.2023.40, author = {Holm, Jacob and van der Hoog, Ivor and Rotenberg, Eva}, title = {{Worst-Case Deterministic Fully-Dynamic Biconnectivity in Changeable Planar Embeddings}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {40:1--40:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.40}, URN = {urn:nbn:de:0030-drops-178909}, doi = {10.4230/LIPIcs.SoCG.2023.40}, annote = {Keywords: dynamic graphs, planarity, connectivity} }
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