Upward planar drawings with two slopes

TL;DR

This paper studies upward planar digraph drawings using only two slopes, providing algorithms for fixed and variable embedding scenarios, and applying the results to phylogenetic networks.

Contribution

It introduces linear-time algorithms for upward 2-slope drawings in fixed embedding and single-source cases, and fixed-parameter algorithms for general digraphs.

Findings

Linear-time construction for fixed embedding scenario.

Algorithms for single-source and series-parallel digraphs.

Application to upward planar phylogenetic networks with leaves on a line.

Abstract

In an upward planar 2-slope drawing of a digraph, edges are drawn as straight-line segments in the upward direction without crossings using only two different slopes. We investigate whether a given upward planar digraph admits such a drawing and, if so, how to construct it. For the fixed embedding scenario, we give a simple characterisation and a linear-time construction by adopting algorithms from orthogonal drawings. For the variable embedding scenario, we describe a linear-time algorithm for single-source digraphs, a quartic-time algorithm for series-parallel digraphs, and a fixed-parameter tractable algorithm for general digraphs. For the latter two classes, we make use of SPQR-trees and the notion of upward spirality. As an application of this drawing style, we show how to draw an upward planar phylogenetic network with two slopes such that all leaves lie on a horizontal line.