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.
Highlights
Two fundamental primitives for highlighting structural properties of a graph in a drawing are alignment of vertices such that they are collinear and geometrically separating unrelated graph parts, e.g., separating them by a straight line
We show that the problem is N P-hard but fixed-parameter tractable (FPT) with respect to |S|
We show that every aligned graph where each edge either entirely lies on a pseudoline or is intersected by at most one pseudoline, i.e., alignment complexity (1, 0, ⊥), has an aligned drawing
Summary
Two fundamental primitives for highlighting structural properties of a graph in a drawing are alignment of vertices such that they are collinear and geometrically separating unrelated graph parts, e.g., separating them by a straight line. Da Lozzo et al [5] show that this is a full characterization of the alignment problem, i.e., a straight-line drawing where the vertices in S are collinear exists if and only if there exists a pseudoline L with respect to G that contains the vertices in S. The results of Da Lozzo et al [5] show that given a pseudoline L with respect to G one can always find a planar straight-line drawing of G such that the vertices on L are collinear and the vertices contained in the halfplanes defined by L are separated by a line L. Chaplik et al [3] study the problem of drawing planar graphs such that all edges can be covered by k lines They show that it is N P-hard to decide whether such a drawing exists. The proofs of statements marked with ( ) can be found in the appendix
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
