Let G be a graph embedded in the plane and let A be an arrangement of pseudolines intersecting the drawing of G.  aligned drawing of G and 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 A. We show that if A is stretchable and every edge e either entirely lies on a pseudoline or intersects at most one pseudoline, then G and 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 C 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{P -hard but fixed-parameter tractable.
Let G be a graph embedded in the plane and let A be an arrangement of pseudolines intersecting the drawing of G. An aligned drawing of G and 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 A. We show that if A is stretchable and every edge e either entirely lies on a pseudoline or intersects at most one pseudoline, then G and 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 C 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 -hard but fixed-parameter tractable.
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\mathcal{P
\mathcal{NP