On Obstacle Numbers

TL;DR

This paper improves the lower bound on the obstacle number of graphs, showing that some graphs require at least Omega(n/( extlog extlog n)^2) obstacles, advancing understanding of this graph parameter.

Contribution

It enhances the lower bound for obstacle numbers in graphs from Omega(n/ extlog n) to Omega(n/( extlog extlog n)^2), using bounds on graph counts.

Findings

Lower bound on obstacle number increased to Omega(n/( extlog extlog n)^2)

Method relies on bounds of graphs with obstacle number at most h

Any improvements in upper bounds will further improve the lower bound

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.