On Obstacle Numbers

Abstract: The obstacle number is a new graph parameter introduced by Alpert, Koch, and Laison (2010). Mukkamala etal (2012) show that there exist graphs with n vertices having obstacle number in Omega(n/\log n). In this note, we up this lower bound to Omega(n/(\log\log n)^2. Our proof makes use of an upper bound of Mukkamala etal on the number of graphs having obstacle number at most h in such a way that any subsequent improvements to their upper bound will improve our lower bound.