  We present an algorithm to support the dynamic embedding in the plane of a
dynamic graph. An edge can be inserted across a face between two vertices on
the face boundary (we call such a vertex pair linkable), and edges can be
deleted. The planar embedding can also be changed locally by flipping
components that are connected to the rest of the graph by at most two vertices.
  Given vertices $u,v$, linkable$(u,v)$ decides whether $u$ and $v$ are
linkable in the current embedding, and if so, returns a list of suggestions for
the placement of $(u,v)$ in the embedding. For non-linkable vertices $u,v$, we
define a new query, one-flip-linkable$(u,v)$ providing a suggestion for a flip
that will make them linkable if one exists. We support all updates and queries
in O(log$^2 n$) time. Our time bounds match those of Italiano et al. for a
static (flipless) embedding of a dynamic graph.
  Our new algorithm is simpler, exploiting that the complement of a spanning
tree of a connected plane graph is a spanning tree of the dual graph. The
primal and dual trees are interpreted as having the same Euler tour, and a main
idea of the new algorithm is an elegant interaction between top trees over the
two trees via their common Euler tour.
