We show that if a planar graph G has a plane straight-line drawing in which a subset S of its vertices are collinear, then for any set of points, X, in the plane with |X| = |S|, there is a plane straight-line drawing of G in which the vertices in S are mapped to the points in X. This solves an open problem posed by Ravsky and Verbitsky in 2008. In their terminology, we show that every collinear set is free. This result has applications in graph drawing, including untangling, column planarity, universal point subsets, and partial simultaneous drawings.

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Conference 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019
Citation
Dujmović, V, Frati, F. (Fabrizio), Gonçalves, D. (Daniel), Morin, P, & Rote, G. (Günter). (2019). Every collinear set in a planar graph is free. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 1521–1538).