  A directed graph $G$ is upward planar if it admits a planar embedding such
that each edge is $y$-monotone. Unlike planarity testing, upward planarity
testing is NP-hard except in restricted cases, such as when the graph has the
single-source property (i.e. each connected component only has one source).
  In this paper, we present a dynamic algorithm for maintaining a combinatorial
embedding $\mathcal{E}(G)$ of a single-source upward planar graph subject to
edge deletions, edge contractions, edge insertions upwards across a face, and
single-source-preserving vertex splits through specified corners. We
furthermore support changes to the embedding $\mathcal{E}(G)$ on the form of
subgraph flips that mirror or slide the placement of a subgraph that is
connected to the rest of the graph via at most two vertices.
  All update operations are supported as long as the graph remains upward
planar, and all queries are supported as long as the graph remains
single-source. Updates that violate upward planarity are identified as such and
rejected by our update algorithm. We dynamically maintain a linear-size data
structure on $G$ which supports incidence queries between a vertex and a face,
and upward-linkability of vertex pairs. If a pair of vertices are not
upwards-linkable, we facilitate one-flip-linkable queries that point to a
subgraph flip that makes them linkable, if any such flip exists.
  We support all updates and queries in $O(\log^2 n)$ time.
