Aligned Drawings of Planar Graphs - Mchedlidze, Tamara; Radermacher, Marcel; Rutter, Ignaz (Journal of Graph Algorithms and Applications, 2018)
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 $Γ$ of $G$ with an arrangement $A$ of lines so that $Γ$ 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. [Da Lozzo et al. GD 2016], 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 $\mathcal{NP}$-complete but fixed-parameter tractable.