First order limits of sparse graphs: Plane trees and path-width
Supported by Czech Science Foundation (to J.G., P.H., and J.O.) (Project 14-03501S); European Social Fund (to S.O.); The State Budget of Czech Republic (to S.O.) [CZ.1.07/2.3.00/30.0009 (POSTDOC I)]; Czech Science foundation (to T.K.) (Project GA14-19503S); European Research Council (European Union's Seventh Framework Programme FP7/2007-2013)/ERC (to D.K.) (259385); Engineering and Physical Sciences Research Council Standared Grant (to D.K.) (EP/M025365/1).
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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
