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 $${{\mathrm{obs}}}(G) \le 2n\log {n}$$ . This refutes a conjecture of Mukkamala, Pach, and Pálvölgyi. 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 $$2^{\varOmega (hn)}$$ on the number of n-vertex graphs with obstacle number at most h for $$h<n$$ and an asymptotically matching lower bound $$\varOmega (n^{4/3}M^{2/3})$$ for the complexity of a collection of $$M \ge \varOmega (n)$$ faces in an arrangement of $$n^2$$ line segments with 2n endpoints.