  The slope number of a graph $G$ is the smallest number of slopes needed for
the segments representing the edges in any straight-line drawing of $G$. It
serves as a measure of the visual complexity of a graph drawing. Several bounds
on the slope number for particular graph classes have been established, both in
the planar and the non-planar setting. Moreover, the slope number can also be
defined for directed graphs and upward planar drawings.
  We study upward planar straight-line drawings that use only a constant number
of slopes. In particular, for a fixed number $k$ of slopes, we are interested
in whether a given directed graph $G$ with maximum in- and outdegree at most
$k$ admits an upward planar $k$-slope drawing. We investigate this question
both in the fixed and the variable embedding scenario. We show that this
problem is in general NP-hard to decide for outerplanar graphs ($k = 3$) and
planar graphs ($k \ge 3$). On the positive side, we can decide whether a given
cactus graph admits an upward planar $k$-slope drawing and, in the affirmative,
construct such a drawing in FPT time with parameter $k$. Furthermore, we can
determine the minimum number of slopes required for a given tree in linear time
and compute the corresponding drawing efficiently.
