Lower bounds on the obstacle number of graphs

TL;DR

This paper establishes lower bounds on the obstacle number of graphs, showing that some graphs require at least on the order of n divided by log n obstacles in their obstacle representations.

Contribution

It proves that there exist graphs with obstacle number at least proportional to n divided by log n, providing a significant lower bound in the study of obstacle representations.

Findings

Existence of graphs with obstacle number at least Ω(n / log n)

Lower bounds on obstacle number for certain graphs

Advancement in understanding obstacle representations of graphs

Abstract

Given a graph $G$, an {\em obstacle representation} of $G$ is a set of points in the plane representing the vertices of $G$, together with a set of connected obstacles such that two vertices of $G$ are joined by an edge if and only if the corresponding points can be connected by a segment which avoids all obstacles. The {\em obstacle number} of $G$ is the minimum number of obstacles in an obstacle representation of $G$. It is shown that there are graphs on $n$ vertices with obstacle number at least $\Omega({n}/{\log n})$.