Let's consider a graph $G(n)$ of $n$ vertices such as no two vertices in $G$ have the same exact neighbors (different open neighbourhoods to be more specific; I wonder if this kind of graphs have already a name). Which is the maximum number of maximal cliques of such graphs? Is that amount polynomically bounded?
Additional related question: which extra conditions must such graphs have to be planar? This is in fact a related question because planar graph does indeed have a polynomically bounded amount of maximal cliques.