Exact neighbourhood complexity for bounded-treewidth graphs

Determine, up to a constant factor, the neighbourhood complexity of graphs with bounded treewidth.

Background

Theorem~\ref{thm:TW} establishes an O_t(r^{t+1}k) upper bound for the r-profile complexity of graphs of treewidth at most t, which also implies an O_t(r^{t+1}k) upper bound for r-neighbourhood complexity. A construction cited from Joret and Rambaud gives a lower bound of Ω_t(r^t|A|) for neighbourhood complexity, leaving a gap in the exponent of r. The authors therefore ask for the optimal neighbourhood-complexity bound up to a constant factor.