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Special issue on Selected papers from the Twentyfifth International Symposium on Graph Drawing and Network Visualization, GD 2017
DOI: 10.7155/jgaa.00475
Aligned Drawings of Planar Graphs
Vol. 22, no. 3, pp. 401429, 2018. Regular paper.
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
straightline 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 fixedparameter tractable.

Submitted: November 2017.
Reviewed: April 2018.
Revised: May 2018.
Reviewed: June 2018.
Revised: July 2018.
Accepted: July 2018.
Final: July 2018.
Published: September 2018.

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