2016/10/15 by Balko, Martin, Cibulka, Josef, Valtr, Pavel · 1 citation
#05C62 #68R10 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1610.04741
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 \rm 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 \rm obs(G) ≤ n\lceillogn\rceil-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 2Ω(hn) on the number of n-vertex graphs with obstacle number at most h for h