Journal of Symbolic Logic

First order properties on nowhere dense structures

Jaroslav Nešetřil and Patrice Ossona de Mendez

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
J. Symbolic Logic, Volume 75, Issue 3 (2010), 868-887.

Dates
First available in Project Euclid: 9 July 2010

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1278682204

Digital Object Identifier
doi:10.2178/jsl/1278682204

Mathematical Reviews number (MathSciNet)
MR2723771

Zentralblatt MATH identifier
1206.03033

Citation

Nešetřil, Jaroslav; Ossona de Mendez, Patrice. First order properties on nowhere dense structures. J. Symbolic Logic 75 (2010), no. 3, 868--887. doi:10.2178/jsl/1278682204. https://projecteuclid.org/euclid.jsl/1278682204


Export citation