On the Number of Minimal Separators in Graphs

Abstract: We consider the largest number of minimal separators a graph on n vertices can have at most. We give a new proof that this number is in $O( ((1+\sqrt{5})/2)^n n )$. We prove that this number is in $\omega( 1.4521^n )$, improving on the previous best lower bound of $\Omega(3^{n/3}) \subseteq \omega( 1.4422^n )$. This gives also an improved lower bound on the number of potential maximal cliques in a graph. We would like to emphasize that our proofs are short, simple, and elementary.