First order limits of sparse graphs: Plane trees and path‐width

Nešetřil and Ossona de Mendez introduced the notion of first order convergence as an attempt to unify the notions of convergence for sparse and dense graphs. It is known that there exist first order convergent sequences of graphs with no limit modeling (an analytic representation of the limit). On t...

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Bibliographic Details
Published in:Random structures & algorithms 2017-07, Vol.50 (4), p.612-635
Main Authors: Gajarský, Jakub, Hliněný, Petr, Kaiser, Tomáš, Král’, Daniel, Kupec, Martin, Obdržálek, Jan, Ordyniak, Sebastian, Tůma, Vojtěch
Format: Article
Language:English
Subjects:
Convergence
first order limits
first order logic
graph limits
Graphs
graphs with bounded path‐width
Modelling
Sequences
Trees
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Summary:Nešetřil and Ossona de Mendez introduced the notion of first order convergence as an attempt to unify the notions of convergence for sparse and dense graphs. It is known that there exist first order convergent sequences of graphs with no limit modeling (an analytic representation of the limit). On the positive side, every first order convergent sequence of trees or graphs with no long path (graphs with bounded tree‐depth) has a limit modeling. We strengthen these results by showing that every first order convergent sequence of plane trees (trees with embeddings in the plane) and every first order convergent sequence of graphs with bounded path‐width has a limit modeling. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 612–635, 2017
ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.20676