  We study monotone simultaneous embeddings of upward planar digraphs, which
are simultaneous embeddings where the drawing of each digraph is upward planar,
and the directions of the upwardness of different graphs can differ. We first
consider the special case where each digraph is a directed path. In contrast to
the known result that any two directed paths admit a monotone simultaneous
embedding, there exist examples of three paths that do not admit such an
embedding for any possible choice of directions of monotonicity. We prove that
if a monotone simultaneous embedding of three paths exists then it also exists
for any possible choice of directions of monotonicity. We provide a
polynomial-time algorithm that, given three paths, decides whether a monotone
simultaneous embedding exists and, in the case of existence, also constructs
such an embedding. On the other hand, we show that already for three paths, any
monotone simultaneous embedding might need a grid of exponential (w.r.t. the
number of vertices) size. For more than three paths, we present a
polynomial-time algorithm that, given any number of paths and predefined
directions of monotonicity, decides whether the paths admit a monotone
simultaneous embedding with respect to the given directions, including the
construction of a solution if it exists. Further, we show several implications
of our results on monotone simultaneous embeddings of general upward planar
digraphs. Finally, we discuss complexity issues related to our problems.
