The number of labeled graphs of bounded treewidth

TL;DR

This paper derives bounds on the number of labeled graphs with a given number of vertices and bounded treewidth, extending previous specific cases to a general formula with implications for related graph classes.

Contribution

It provides explicit upper and lower bounds for counting labeled graphs of bounded treewidth, including new algorithmic construction-based lower bounds.

Findings

Established bounds for T_{n,k} for all k > 1

Bounds also apply to graphs with bounded pathwidth and proper-pathwidth

The upper bound follows from known counts of labeled k-trees

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