First order limits of sparse graphs: Plane trees and path‐width. Issue 4 (30th August 2016)

• Record Type: Journal Article
• Authors: Gajarský, Jakub; Hliněný, Petr; Kaiser, Tomáš; Král', Daniel; Kupec, Martin; Obdržálek, Jan; Ordyniak, Sebastian; Tůma, Vojtěch
• Is Part Of: Random structures & algorithms. Volume 50:Issue 4(2017)
• Page Start: 612
• Page End: 635
• Publication Date: 2016-08-30
• Subjects: graph limits -- graphs with bounded path‐width -- first order limits -- first order logic; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20676
• Languages: English
• ISSNs: 1042-9832

Abstract

Abstract: 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