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Nesetril and Ossona de Mendez introduced the notion of first orderconvergence as an attempt to unify the notions of convergence for sparse anddense graphs. It is known that there exist first order convergent sequences ofgraphs with no limit modeling (an analytic representation of the limit). On thepositive side, every first order convergent sequence of trees or graphs with nolong path (graphs with bounded tree-depth) has a limit modeling. We strengthenthese results by showing that every first order convergent sequence of planetrees (trees with embeddings in the plane) and every first order convergentsequence of graphs with bounded path-width has a limit modeling.

First order limits of sparse graphs: Plane trees and path-width