Drawing graphs using a small number of obstacles

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.