Rooting algebraic vertices of convergent sequences

0573756 - ÚI 2024 RIV CZ eng C - Conference Paper (international conference)
Hartman, David - Hons, T. - Nešetřil, J.
Rooting algebraic vertices of convergent sequences.
EUROCOMB’23. Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications. Brno: MUNI Press, 2023 - (Kráľ, D.; Nešetřil, J.), s. 539-544. E-ISSN 2788-3116.
[EUROCOMB 2023: European Conference on Combinatorics, Graph Theory and Applications /12./. Prague (CZ), 28.08.2023-01.09.2023]
Institutional support: RVO:67985807
Keywords : rooting * algebraic vertices * convergent sequences
OECD category: Pure mathematics
https://journals.phil.muni.cz/eurocomb/article/view/35609/31523

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 (Gn) converging to a limit L and a vertex r of L it is possible to find a sequence of vertices (rn) such that L rooted at r is the limit of the graphs Gn rooted at rn. A counterexample was found by Christofides and Král’, 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 (rn) always exists.
Permanent Link: https://hdl.handle.net/11104/0344128