  In this paper we study the computational complexity of the Upward Planarity
Extension problem, which takes in input an upward planar drawing $\Gamma_H$ of
a subgraph $H$ of a directed graph $G$ and asks whether $\Gamma_H$ can be
extended to an upward planar drawing of $G$. Our study fits into the line of
research on the extensibility of partial representations, which has recently
become a mainstream in Graph Drawing.
  We show the following results.
  First, we prove that the Upward Planarity Extension problem is NP-complete,
even if $G$ has a prescribed upward embedding, the vertex set of $H$ coincides
with the one of $G$, and $H$ contains no edge.
  Second, we show that the Upward Planarity Extension problem can be solved in
$O(n \log n)$ time if $G$ is an $n$-vertex upward planar $st$-graph. This
result improves upon a known $O(n^2)$-time algorithm, which however applies to
all $n$-vertex single-source upward planar graphs.
  Finally, we show how to solve in polynomial time a surprisingly difficult
version of the Upward Planarity Extension problem, in which $G$ is a directed
path or cycle with a prescribed upward embedding, $H$ contains no edges, and no
two vertices share the same $y$-coordinate in $\Gamma_H$.
