Vida Dujmović, Fabrizio Frati, Daniel Gonçalves, Pat Morin, and Günter Rote:

Every collinear set in a planar graph is free

1. In: Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA19), San Diego, January 2019, Editor: Timothy Chan. pp. 1521–1538, SIAM. doi:10.1137/1.9781611975482.92 →BibTeX
2. Discrete and Computational Geometry 65 (2021), pp. 999–1027, doi:10.1007/s00454-019-00167-x arXiv:1811.03432 [math.CO]. →BibTeX

Abstract

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 X of points 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.