On preservation under homomorphisms and unions of conjunctive queries

Unions of conjunctive queries, also known as select-project-join-union queries, are the most frequently asked queries in relational database systems. These queries are definable by existential positive first-order formulas and are preserved under homomorphisms. A classical result of mathematical logic asserts that the existential positive formulas are the only first-order formulas (up to logical equivalence) that are preserved under homomorphisms on all structures, finite and infinite. The question of whether the homomorphism-preservation theorem holds for the class of all finite structures resisted solution for a long time. It was eventually shown that, unlike other classical preservation theorems, the homomorphism-preservation theorem does hold in the finite. In this article, we show that the homomorphism-preservation theorem holds also for several restricted classes of finite structures of interest in graph theory and database theory. Specifically, we show that this result holds for all classes of finite structures of bounded degree, all classes of finite structures of bounded treewidth, and, more generally, all classes of finite structures whose cores exclude at least one minor.