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.

Collinear sets, Graph drawing, Planar graphs, Untangling
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