Aligned Drawings of Planar Graphs

TL;DR

This paper investigates conditions under which planar graphs can be drawn with vertices and edges aligned to pseudoline arrangements, providing new existence results and complexity insights for related alignment problems.

Contribution

It proves that certain aligned drawings exist for stretchable pseudoline arrangements and strengthens existing results, also analyzing the complexity of vertex collinearity problems.

Findings

Existence of straight-line aligned drawings under specific conditions

Strengthening of a previous result on convex drawings with a pseudoline

NP-completeness of the vertex collinearity problem

Abstract

Let $G$ be a graph that is topologically embedded in the plane and let $\mathcal{A}$ be an arrangement of pseudolines intersecting the drawing of $G$. An aligned drawing of $G$ and $\mathcal{A}$ is a planar polyline drawing $\Gamma$ of $G$ with an arrangement $A$ of lines so that $\Gamma$ and $A$ are homeomorphic to $G$ and $\mathcal{A}$. We show that if $\mathcal{A}$ is stretchable and every edge $e$ either entirely lies on a pseudoline or it has at most one intersection with $\mathcal{A}$, then $G$ and $\mathcal{A}$ have a straight-line aligned drawing. In order to prove this result, we strengthen a result of Da Lozzo et al., and prove that a planar graph $G$ and a single pseudoline $\mathcal{L}$ have an aligned drawing with a prescribed convex drawing of the outer face. We also study the less restrictive version of the alignment problem with respect to one line, where only a set of vertices is given and we need to determine whether they can be collinear. We show that the problem is NP-complete but fixed-parameter tractable.