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: Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science, arXiv:0806.0253). 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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doi.org/10.1007/s00454-019-00167-x
Discrete and Computational Geometry
School of Computer Science

Dujmović, V, Frati, F. (Fabrizio), Gonçalves, D. (Daniel), Morin, P, & Rote, G. (Günter). (2020). Every Collinear Set in a Planar Graph is Free. Discrete and Computational Geometry. doi:10.1007/s00454-019-00167-x