The first-order-transduction characterization of bounded merge-width

For a graph class $\mathcal{C}$, say that it has bounded merge-width when its merge-width is bounded by a constant. A first-order transduction is an interpretation of graphs from structures using first-order formulas, and a graph class has linear neighbourhood complexity when the number of distinct neighbourhoods induced on every $p$-element vertex set is $O(p)$. First-order-transduction characterization of bounded merge-width. A class $\mathcal{C}$ has bounded merge-width if and only if every first-order transduction of $\mathcal{C}$ has linear neighbourhood complexity. This proposes a dense-graph analogue of the characterization of bounded expansion by linear neighbourhood complexity, replacing ordinary neighbourhoods by first-order transductions. The paper establishes linear neighbourhood complexity for bounded merge-width at radius $2$, but the stated if-and-only-if characterization is not proved in the supplied text.