Number of maximal cliques in graphs without common neighbourhoods 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.
Suppose that $n$ is a perfect square, and identify its vertex set with $\{1,\ldots,\sqrt{n}\}^2$. Connect $(a,b)$ to $(c,d)$ if $a \neq c$ and $b \neq d$. You can identify a vertex from the set of its neighbors, so the open neighborhoods are all distinct. The number of maximal cliques is $\sqrt{n}!$, which isn't polynomial.