Abstrakt
Structural convergence is a framework for convergence of graphs by Nešetřil 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 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 Král', but they showed that the statement holds for almost all vertices 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 is an algebraic vertex (i.e. belongs to a finite definable set), the sequence of roots always exists.