Upward Planar Drawings with Three and More Slopes

TL;DR

This paper investigates the complexity of creating upward planar straight-line graph drawings with a limited number of slopes, establishing NP-hardness results and providing efficient algorithms for specific graph classes.

Contribution

It proves NP-hardness of the slope-limited upward planar drawing problem for certain graph classes and offers fixed-parameter tractable algorithms for cactus graphs and linear-time solutions for trees.

Findings

NP-hardness for outerplanar and planar graphs with 3 or more slopes

FPT algorithm for cactus graphs

Linear-time algorithm for trees

Abstract

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.