The number of labeled graphs of bounded treewidth

We focus on counting the number of labeled graphs on $n$ vertices and treewidth at most $k$ (or equivalently, the number of labeled partial $k$-trees), which we denote by $T_{n,k}$. So far, only the particular cases $T_{n,1}$ and $T_{n,2}$ had been studied. We show that $$ left(c cdot frac{kcdot 2^k cdot n}{log k} right)^n cdot 2^{-frac{k(k+3)}{2}} cdot k^{-2k-2} leq T_{n,k} leq left(k cdot 2^k cdot nright)^n cdot 2^{-frac{k(k+1)}{2}} cdot k^{-k}, $$ for $k > 1$ and some explicit absolute constant $c > 0$. The upper bound is an immediate consequence of the well-known number of labeled $k$-trees, while the lower bound is obtained from an explicit algorithmic construction. It follows from this construction that both bounds also apply to graphs of pathwidth and proper-pathwidth at most $k$.