Maximum cliques of the transitive closure of a chordal DAG

Question

Let $G=(V,A)$ be a directed acyclic graph, for which the underlying undirected graph is chordal (so that every induced cycle in the underlying undirected graph is a triangle).

It is known that in a chordal graph the number of maximum cliques is linearly bounded in the number $|V|$ of vertices.

Let $G'=(V,A')$ be the transitive closure of $G$, so that for every directed path $p= (v_i,\ldots,v_j)$ from $v_i$ to $v_j$ in $G$, there exists a directed edge $(v_i,v_j)$ in $A'$.

Question: Is there any polynomial/linear bound on the number of maximum cliques in the undirected graph that underlies $G'$?

Answer 1

I think you can extend Vinicius dos Santos' idea to show that no polynomial bound is possible.

Consider a graph on $n$ vertices divided into $d\geq 1$ groups of size about $n/d$ as follows:

Its transitive closure has about $(\frac{n}{d})^d$ maximal (undirected) cliques.

Answer 2

A linear bound is impossible. Suppose your graph is a star, with half of the edges oriented towards the universal vertex $u$ and the rest oriented outward. In the transitive closure, the maximal cliques will be triangles, each of them containing $u$ and exactly one of its inneighbors and one of its outneighbors. This gives a quadratic number of maximal (and maximum) cliques.