  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.
