Structural convergence and algebraic roots

TL;DR

This paper investigates the conditions under which a sequence of rooted graphs converges to a limit rooted at a specific vertex, focusing on algebraic vertices and their definable sets within the framework of structural convergence.

Contribution

It introduces the concept of algebraic vertices and proves that for such vertices, the sequence of roots converging to a limit rooted graph always exists.

Findings

Existence of rooted graph sequences for algebraic vertices

Counterexamples for non-algebraic vertices

Extension of structural convergence theory

Abstract

Structural convergence is a framework for convergence of graphs by Ne\v{s}et\v{r}il and Ossona de Mendez that unifies the dense (left) graph convergence and Benjamini-Schramm convergence. They posed a problem asking whether for a given sequence of graphs $(G_n)$ converging to a limit $L$ and a vertex $r$ of $L$ it is possible to find a sequence of vertices $(r_n)$ such that $L$ rooted at $r$ is the limit of the graphs $G_n$ rooted at $r_n$. A counterexample was found by Christofides and Kr\'{a}l', but they showed that the statement holds for almost all vertices $r$ of $L$. We offer another perspective to the original problem by considering the size of definable sets to which the root $r$ belongs. We prove that if $r$ is an algebraic vertex (i.e. belongs to a finite definable set), the sequence of roots $(r_n)$ always exists.