Strong modeling limits of graphs with bounded tree-width

TL;DR

This paper proves that for sequences of graphs with bounded tree-width, there exist modeling limits that satisfy the strong finitary mass transport principle, extending previous results to a broader class of graphs.

Contribution

It establishes the existence of such modeling limits for graphs with bounded tree-width, filling a gap in the understanding of graph convergence properties.

Findings

Existence of modeling limits for bounded tree-width graphs.

Satisfaction of the strong finitary mass transport principle.

Extends previous results from trees and path-width to bounded tree-width graphs.

Abstract

The notion of first order convergence of graphs unifies the notions of convergence for sparse and dense graphs. Ne\v{s}et\v{r}il and Ossona de Mendez [J. Symbolic Logic 84 (2019), 452-472] proved that every first order convergent sequence of graphs from a nowhere-dense class of graphs has a modeling limit and conjectured the existence of such modeling limits with an additional property, the strong finitary mass transport principle. The existence of modeling limits satisfying the strong finitary mass transport principle was proved for first order convergent sequences of trees by Ne\v{s}et\v{r}il and Ossona de Mendez [Electron. J. Combin. 23 (2016), P2.52] and for first order sequences of graphs with bounded path-width by Gajarsk\'y et al. [Random Structures Algorithms 50 (2017), 612-635]. We establish the existence of modeling limits satisfying the strong finitary mass transport principle for first order convergent sequences of graphs with bounded tree-width.