Homomorphism is a key notion for studying algebraic or relational structures, and many combinatorial problems can be reduced to homomorphism problems. In this chapter, it is shown that computing BG homomorphisms between two BGs is “strongly equivalent” to important combinatorial problems in graph theory, algebra, database theory, and constraint satisfaction networks. In Sect. 5.1, basic
... [Show full abstract] conceptual hypergraphs (BHs) are introduced, with a BH homomorphism notion, while highlighting relationships between BG homomorphisms and BH homomorphisms. The two notions are so close that they can be simply considered as two different views of the same abstract notion. It is also shown in this section that computing homomorphims for BGs on any vocabulary is equivalent to computing homo-morphisms for BGs on specific vocabularies. BGs are kinds of graphs, thus relationships between graph homomorphisms and BG homomorphisms are studied in Sect. 5.2. Section 5.3 concerns relational structures and relational databases. It is especially shown that two fundamental problems dealing with conjunctive, positive and non-recursive queries—the query evaluation problem and the query containment problem—are equivalent to the BG homomorphism problem. Section 5.4 is devoted to the Constraint Satisfaction Problem (CSP), which is the basic problem in the constraint processing domain, and its equivalence to the BG homomorphism problem is stated. This equivalence, that borrows a lot from techniques developed in the CSP framework, will be used in Chap. 6.
