Equivalent definitions of recognizability for sets of graphs of bounded tree-width
Bruno Courcelle, Jens Lagergren · Mathematical Structures in Computer Science · 1996
We show that a set of finite graphs of tree-width at most k is recognizable (with respect to the algebra of graphs with an unbounded number of sources) if and only if it is recognizable with respect to the algebra of graphs of tree-width at most k with at most k sources.