On Collinear Sets in Straight-Line Drawings

Abstract

We consider straight-line drawings of a planar graph G with possible edge crossings. The untangling problem is to eliminate all edge crossings by moving as few vertices as possible to new positions. Let fixG denote the maximum number of vertices that can be left fixed in the worst case among all drawings of G. In the allocation problem, we are given a planar graph G on n vertices together with an n-point set X in the plane and have to draw G without edge crossings so that as many vertices as possible are located in X. Let fitG denote the maximum number of points fitting this purpose in the worst case among all n-point sets X. As fixG≤fitG, we are interested in upper bounds for the latter and lower bounds for the former parameter. For any e>0, we construct an infinite sequence of graphs with fitG=O(nσ+e), where σ $fix G\ge\sqrt{n/30}$ for any graph G of tree-width at most 2. This extends the lower bound obtained by Goaoc et al. [Discrete and Computational Geometry 42:542---569 (2009)] for outerplanar graphs. Our results are based on estimating the maximum number of vertices that can be put on a line in a straight-line crossing-free drawing of a given planar graph.