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Monotone Simultaneous Embedding of Directed Paths

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.