Monotone Simultaneous Embedding of Directed Paths

TL;DR

This paper investigates the conditions and algorithms for monotone simultaneous embeddings of directed paths, providing polynomial-time decision procedures and exploring the complexity and size bounds of such embeddings.

Contribution

It introduces polynomial-time algorithms for deciding and constructing monotone simultaneous embeddings of multiple directed paths, and analyzes their size and complexity bounds.

Findings

Decidable in polynomial time whether three paths admit a monotone simultaneous embedding.

Existence of embeddings may require exponential grid size.

Algorithms extend to multiple paths with predefined directions and implications for general upward planar digraphs.

Abstract

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.