On collinear sets in straight line drawings

TL;DR

This paper investigates the limits of straight line drawings of planar graphs with crossings, establishing new bounds on the number of vertices that can be fixed or fit in specific configurations, with implications for untangling and allocation problems.

Contribution

It provides the first examples of graphs with sublinear fit in point sets and extends lower bounds for fixed vertices in graphs of tree-width two.

Findings

Constructed graphs with fit(G)=O(n^{σ+ε}) for σ<0.99

Proved fix(G)≥√(n/30) for graphs with tree-width ≤ 2

Graphs of tree-width 2 can have large collinear vertex sets in crossing-free drawings

Abstract

We consider straight line drawings of a planar graph $G$ with possible edge crossings. The \emph{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. In the \emph{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. As $fix(G)\le fit(G)$, we are interested in upper bounds for the latter and lower bounds for the former parameter. For each $\epsilon>0$, we construct an infinite sequence of graphs with $fit(G)=O(n^{\sigma+\epsilon})$, where $\sigma<0.99$ is a known graph-theoretic constant, namely the shortness exponent for the class of cubic polyhedral graphs. To the best of our knowledge, this is the first example of graphs with $fit(G)=o(n)$. On the other hand, we prove that $fix(G)\ge\sqrt{n/30}$ for all $G$ with 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 upper bound for $fit(G)$ is based on the fact that the constructed graphs can have only few collinear vertices in any crossing-free drawing. To prove the lower bound for $fix(G)$, we show that graphs of tree-width 2 admit drawings that have large sets of collinear vertices with some additional special properties.