The number of labeled graphs of bounded treewidth
Julien Baste, Marc Noy, Ignasi Sau

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 vertices and treewidth at most (or equivalently, the number of labeled partial -trees), which we denote by . So far, only the particular cases and had been studied. We show that for and some explicit absolute constant . The upper bound is an immediate consequence of the well-known number of labeled -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 .
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