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.

翻译：结构收敛性是Nešetřil和Ossona de Mendez提出的图收敛框架，统一了稠密（左）图收敛与Benjamini-Schramm收敛。他们提出如下问题：给定收敛于极限L的图序列(G_n)及L的顶点r，是否存在顶点序列(r_n)使得以r为根节点的L成为以r_n为根节点的图序列G_n的极限？Christofides和Král'给出了反例，但他们证明该结论对L中几乎所有顶点r成立。我们从r所属的可定义集大小角度提出原问题的新视角，证明若r为代数顶点（即属于有限可定义集），则根节点序列(r_n)必然存在。