The number of labeled graphs of bounded treewidth

Abstract

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 $${\left(c\cdot \frac{k\cdot 2^k\cdot n}{\log k}\right)}^n\cdot 2^{-\frac{k(k+3)}{2}}\cdot k^{-2k-2}\le T_{n,k}\le {\left(k\cdot 2^k\cdot n\right)}^n\cdot 2^{-\frac{k(k+1)}{2}}\cdot k^{-k},$$ 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 $k$.

12 pages, 3 figures