A class of structures is said to have the homomorphism-preservation propertyjust in case every first-order formula that is preserved by homomorphisms onthis class is equivalent to an existential-positive formula. It is known by aresult of Rossman that the class of finite structures has this property and byprevious work of Atserias et al. that various of its subclasses do. We extendthe latter results by introducing the notion of a quasi-wide class and showingthat any quasi-wide class that is closed under taking substructures anddisjoint unions has the homomorphism-preservation property. We show, inparticular, that classes of structures of bounded expansion and that locallyexclude minors are quasi-wide. We also construct an example of a class offinite structures which is closed under substructures and disjoint unions butdoes not admit the homomorphism-preservation property.
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