Background
My question is about a generalization of the following situation:
Let $M$ be a finite metric space. Given $r>0$, the $r$-neighborhood graph $N(M)_r$ has vertex set $M$ and an edge $\{x,y\}$ whenever $d_M(x,y) \leq r$.
When $M$ is a tree (equipped with the shortest-path metric), $N(M)_r$ is a chordal graph [1] for any $r >0$, and any chordal graph has at most $|M|$ maximal cliques. Therefore, $N(M)_r$ has at most $|M|$ maximal cliques for any $r$.
Question
Suppose that $M$ is no longer a tree, but instead is a partial $k$-tree, aka a graph of treewidth $k$ for some bounded value of $k$. Is it true that the $r$-neighborhood graphs of $M$ also have a polynomial number of maximal cliques, as a function of $|M|$?
Thoughts
Of course, the same argument as if $M$ is a tree no longer applies. However, I am thinking that perhaps there is some way to approach this question using the tree decomposition of $M$, and then reasoning from there to give a polynomial upper bound on the number of maximal cliques of the neighborhood graphs.
However, it could also be that there is some counterexample that I couldn't think of.
