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First order properties on nowhere dense structures

Published online by Cambridge University Press:  12 March 2014

Jaroslav Nešetřil  and
Patrice Ossona de Mendez
Jaroslav Nešetřil
Affiliation:
Department of Applied Mathematics and Institute of Theoretical Computer Science (ITI), Charles University, Malostranské Nám.25, 11800 Praha 1, Czech Republic. E-mail: [email protected]
Patrice Ossona de Mendez
Affiliation:
Centre d'Analyse et de Mathématiques Sociales CNRS, UMR 8557, 54 BD Raspail, 75006 Paris, France. E-mail: [email protected]
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Abstract

A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Ajtai and Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special classes of structures (such as minor closed classes). This in turn led to the notions of wide, almost wide and quasi-wide classes of graphs. It has been proved previously that minor closed classes and classes of graphs with locally forbidden minors are examples of such classes and thus (relativized) homomorphism preservation theorem holds for them. In this paper we show that (more general) classes with bounded expansion and (newly defined) classes with bounded local expansion and even (very general) nowhere dense classes are quasi wide. This not only strictly generalizes the previous results but it also provides new proofs and algorithms for some of the old results. It appears that bounded expansion and nowhere dense classes are perhaps a proper setting for investigation of wide-type classes as in several instances we obtain a structural characterization. This also puts classes of bounded expansion in the new context. Our motivation stems from finite dualities. As a corollary we obtain that any homomorphism closed first order definable property restricted to a bounded expansion class is a restricted duality.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 3 , September 2010 , pp. 868 - 887
Copyright
Copyright © Association for Symbolic Logic 2010

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