The number of maximal cliques of the intersection graphs

Question

Are there some results about the upper bound of the number of maximal cliques (NMC) of some class of intersection graphs?

I want to know whether some classic classes of intersection graphs have polynomial many of NMC (with respect to $n$, the size of the given graph).

I only know that the NMC of chordal graphs is at most $n$.

I also find some efficient algorithms for listing all the maximal cliques. For example, we can list all maximal cliques on trapezoid graphs in $O(n^{2} + \gamma n)$, where $\gamma$ is the size of the output. However, do we know the trapezoid graphs or permutation graphs have polynomial many NMC?

Answer 1

You might already know about this, but there is a result of Farber (in https://doi.org/10.1016/0012-365X(89)90268-9) that states graphs with no induced $C_4$ have quadratically many maximal cliques.

For permutation graphs, the answer is "no": the complement of $nK_2$ is a permutation graph, and $nK_2$ has $2^n$ maximal stable sets.

Answer 2

One known result due to Brimkov et al. is that intersection graphs of convex polytopes in $\mathbb R^n$ with a bounded number of $(n-1)$-dimensional hyperplanes to which the facets of the polytopes are parallel, have a polynomial number of maximal cliques.

https://www.sciencedirect.com/science/article/pii/S0166218X18301458?via%3Dihub