Lower Bounds on the Obstacle Number of Graphs

Abstract

Given a graph $G$, an 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 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 $\Omega({n}/{\log n})$.