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 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 obs(G) ≤ nlog 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(hn) on the number of n-vertex graphs with obstacle number at most h for h < n and a lower bound (n4/3M2/3) for the complexity of a collection of M ≥ (n log3/2 n) faces in an arrangement of line segments with n endpoints The latter bound is tight up to a multiplicative constant.

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