Drawing Graphs Using a Small Number of Obstacles

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 ${{\mathrm{obs}}}(G)$ 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 ${{\mathrm{obs}}}(G) \le n\lceil \log {n}\rceil -n+1$. This refutes a conjecture of Mukkamala, Pach, and Pálvölgyi. 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 $M \ge \Omega (n\log ^{3/2}{n})$ faces in an arrangement of line segments with n endpoints. The latter bound is tight up to a multiplicative constant.