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 numberobs (G) ofG 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 obs (G) ≤ n⌈ log n⌉ - 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 ^{Ω ( h n )} on the number of n-vertex graphs with obstacle number at most h for h< n and a lower bound Ω (n^{4 / 3}M^{2 / 3}) for the complexity of a collection of M≥ Ω (nlog ^{3 / 2}n) faces in an arrangement of line segments with n endpoints. The latter bound is tight up to a multiplicative constant.