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 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) <= 2n log n. This refutes a conjecture of Mukkamala, Pach, and Palvolgyi. For bipartite n-vertex graphs, we improve this bound to n - 1. 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 an asymptotically matching lower bound Omega(n(4/3) M-2/3) for the complexity of a collection of M >= Omega(n) faces in an arrangement of n(2) line segments with 2n endpoints.