  The Calculus of Conjunctive Queries (CCQ) has foundational status in database
theory. A celebrated theorem of Chandra and Merlin states that CCQ query
inclusion is decidable. Its proof transforms logical formulas to graphs: each
query has a natural model - a kind of graph - and query inclusion reduces to
the existence of a graph homomorphism between natural models.
  We introduce the diagrammatic language Graphical Conjunctive Queries (GCQ)
and show that it has the same expressivity as CCQ. GCQ terms are string
diagrams, and their algebraic structure allows us to derive a sound and
complete axiomatisation of query inclusion, which turns out to be exactly
Carboni and Walters' notion of cartesian bicategory of relations. Our
completeness proof exploits the combinatorial nature of string diagrams as
(certain cospans of) hypergraphs: Chandra and Merlin's insights inspire a
theorem that relates such cospans with spans. Completeness and decidability of
the (in)equational theory of GCQ follow as a corollary. Categorically speaking,
our contribution is a model-theoretic completeness theorem of free cartesian
bicategories (on a relational signature) for the category of sets and
relations.
