Aligned Drawings of Planar Graphs
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Let G be a graph topological 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 Γ of G with an arrangement A of lines so that Γ 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 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 NP-complete but fixed-parameter tractable.
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