Drawing graphs using a small number of obstacles

TL;DR

This paper establishes new upper bounds on the obstacle number of graphs, showing that every n-vertex graph can be represented with a logarithmic number of obstacles, and provides lower bounds on the complexity of obstacle representations.

Contribution

It presents the first non-trivial general upper bound on obstacle numbers and refutes a previous conjecture, also improving bounds for graphs with bounded chromatic number.

Findings

Every n-vertex graph has obstacle number at most n log n - n + 1.

The obstacle number bounds hold even with convex obstacles.

Lower bounds on the number of graphs with small obstacle number and on face complexity in segment arrangements.

Abstract

An obstacle representation of a graph $G$ is a set of points in the plane representing the vertices of $G$, together with a set of polygonal obstacles such that two vertices of $G$ are connected by an edge in $G$ if and only if the line segment between the corresponding points avoids all the obstacles. The obstacle number ${\rm obs}(G)$ of $G$ is the minimum number of obstacles in an obstacle representation of $G$. We provide the first non-trivial general upper bound on the obstacle number of graphs by showing that every $n$-vertex graph $G$ satisfies ${\rm obs}(G) \leq n\lceil\log{n}\rceil-n+1$. This refutes a conjecture of Mukkamala, Pach, and P\'alv\"olgyi. For $n$-vertex graphs with bounded chromatic number, we improve this bound to $O(n)$. Both bounds apply even when the obstacles are required to be convex. We also prove a lower bound $2^{\Omega(hn)}$ on the number of $n$-vertex graphs with obstacle number at most $h$ for $h<n$ and a lower bound $\Omega(n^{4/3}M^{2/3})$ for the complexity of a collection of $M \geq \Omega(n\log^{3/2}{n})$ faces in an arrangement of line segments with $n$ endpoints. The latter bound is tight up to a multiplicative constant.