Every Collinear Set in a Planar Graph Is Free

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 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.

29 pages, 12 figures

References in corpus (2)

• Superpatterns and Universal Point Sets
• Untangling a Planar Graph

Cited by in corpus (1)

• An almost optimal bound on the number of intersections of two simple polygons