Let G be a graph topological 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 intersects at most one pseudoline , then G and \mathcal A have a straight-line aligned drawing. In order to prove these results , we strengthen the 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 more general version of the problem 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 but fixed-parameter tractable.
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.
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