Modeling Limits in Hereditary Classes: Reduction and Application to Trees

TL;DR

This paper explores the concept of modeling limits in hereditary graph classes, particularly focusing on trees, and provides new results supporting the conjecture that nowhere dense classes admit such limits.

Contribution

It extends existing results to show that the class of all finite trees admits modeling limits, advancing the understanding of limits in hereditary graph classes.

Findings

Modeling limits exist for classes of trees with bounded height.

The paper supports the conjecture that nowhere dense classes admit modeling limits.

It develops a general framework for limits in hereditary graph classes.

Abstract

Limits of graphs were initiated recently in the two extreme contexts of dense and bounded degree graphs. This led to elegant limiting structures called graphons and graphings. These approach have been unified and generalized by authors in a more general setting using a combination of analytic tools and model theory to FO-limits (and X-limits) and to the notion of modeling. The existence of modeling limits was established for sequences in a bounded degree class and, in addition, to the case of classes of trees with bounded height and of graphs with bounded tree depth. These seemingly very special classes is in fact a key step in the development of limits for more general situations. The natural obstacle for the existence of modeling limit for a monotone class of graphs is the nowhere dense property and it has been conjectured that this is a sufficient condition. Extending earlier results we derive several general results which present a realistic approach to this conjecture. As an example we then prove that the class of all finite trees admits modeling limits.