Geodesic
Aligned Drawings of Planar Graphs
Journal of Graph Algorithms and Applications, Special issue on Selected papers from the Twenty-fifth International Symposium on Graph Drawing and Network Visualization, GD 2017 , Tome 22 (2018) no. 3, pp. 401-429.

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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. [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.
DOI : 10.7155/jgaa.00475
Keywords: aligned graphs, straight-line drawings, planar graphs, line arrangements, pseudolines 
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     title = {Aligned {Drawings} of {Planar} {Graphs}},
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Tamara Mchedlidze; Marcel Radermacher; Ignaz Rutter. Aligned Drawings of Planar Graphs. Journal of Graph Algorithms and Applications, 
							Special issue on Selected papers from the Twenty-fifth International Symposium on Graph Drawing and Network Visualization, GD 2017
					, Tome 22 (2018) no. 3, pp. 401-429. doi : 10.7155/jgaa.00475. http://geodesic.mathdoc.fr/articles/10.7155/jgaa.00475/

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