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 fix G 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 fit G denote the maximum number of points fitting this purpose in the worst case among all n-point sets X. As fix G ≤ fit G, we are interested in upper bounds for the latter and lower bounds for the former parameter.

For any ε > 0, we construct an infinite sequence of graphs with fit G = O(n ^σ + ε), where σ < 0.99 is a known graph-theoretic constant, namely the shortness exponent for the class of cubic polyhedral graphs. On the other hand, we prove that $fix G\ge\sqrt{n/30}$ 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.