Abstrakt
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) LESS-THAN OR EQUAL TO 2nlogn. This refutes a conjecture of Mukkamala, Pach, and Pálvölgyi.
For bipartite n-vertex graphs, we improve this bound to n MINUS SIGN 1. Both bounds apply even when the obstacles are required to be convex.
We also prove a lower bound 2Ω(hn) on the number of n-vertex graphs with obstacle number at most h for h < M and an asymptotically matching lower bound Ω(n4/3M2/3) for the complexity of a collection of M GREATER-THAN OR EQUAL TO Ω(n) faces in an arrangement of n^2 line segments with 2n endpoints.