Structural convergence and algebraic roots (2310.07045v1)
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.
- “Recurrence of Distributional Limits of Finite Planar Graphs” In Electronic Journal of Probability 6.none Institute of Mathematical Statistics, 2001 DOI: 10.1214/ejp.v6-96
- “Counting Graph Homomorphisms” In Topics in Discrete Mathematics Springer Berlin Heidelberg, 2006, pp. 315–371 DOI: https://doi.org/10.1007/3-540-33700-8˙18
- “First-Order Convergence and Roots” In Combinatorics, Probability and Computing 25.2 Cambridge University Press (CUP), 2015, pp. 213–221 DOI: 10.1017/s0963548315000048
- Gábor Elek “Note on limits of finite graphs” In Combinatorica 27.4 Springer ScienceBusiness Media LLC, 2007, pp. 503–507 DOI: 10.1007/s00493-007-2214-8
- David Hartman, Tomáš Hons and Jaroslav Nešetřil “Gadget construction and structural convergence” arXiv, 2022 DOI: 10.48550/ARXIV.2212.10985
- David Hartman, Tomáš Hons and Jaroslav Nešetřil “Rooting algebraic vertices of convergent sequences” In Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications Masaryk University Press, 2023 DOI: 10.5817/cz.muni.eurocomb23-075
- Wilfrid Hodges “Model Theory” Cambridge University Press, 1993 DOI: 10.1017/cbo9780511551574
- László Lovász “Large networks and graph limits” American Mathematical Society, 2012
- “Limits of dense graph sequences” In Journal of Combinatorial Theory, Series B 96.6 Elsevier BV, 2006, pp. 933–957 DOI: 10.1016/j.jctb.2006.05.002
- Jaroslav Nešetřil and Patrice Ossona Mendez “A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth” In Memoirs of the American Mathematical Society 263.1272 American Mathematical Society (AMS), 2020 DOI: 10.1090/memo/1272
- Jaroslav Nešetřil and Patrice Ossona Mendez “Cluster analysis of local convergent sequences of structures” In Random Structures & Algorithms 51.4 Wiley, 2017, pp. 674–728 DOI: 10.1002/rsa.20719
- Jaroslav Nešetřil and Patrice Ossona Mendez “Existence of Modeling Limits for Sequences of Sparse Structures” In The Journal of Symbolic Logic 84.02 Cambridge University Press (CUP), 2019, pp. 452–472 DOI: 10.1017/jsl.2018.32
- Jaroslav Nešetřil and Patrice Ossona Mendez “Sparsity” Springer Berlin Heidelberg, 2012 DOI: 10.1007/978-3-642-27875-4
- Richard P. Stanley and Sergey Fomin “Enumerative Combinatorics” Cambridge University Press, 1999 DOI: 10.1017/cbo9780511609589
- “A Course in Model Theory” Cambridge University Press, 2012 DOI: 10.1017/cbo9781139015417
- Hassler Whitney “Complex Analytic Varieties” Addsion Weslsey Longman Publishing Group, 1972, pp. 399
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