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We consider the problem of maintaining a dynamic planar graph subject to edge insertions and edge deletions that preserve planarity but that can change the embedding. We describe algorithms and data structures for maintaining information about 2-and 3-vertex-connectivity, and 3-and 4-edge-connectivity in a planar graph in O(n 1/2 ) amortized time per insertion, deletion, or connectivity query. All of the data structures handle insertions that keep the graph planar without regard to anydoi:10.1137/s0097539794269072 fatcat:umm2mcjkbnhv3klkqjyibajese