  Local certification consists in assigning labels (called \emph{certificates})
to the nodes of a network to certify a property of the network or the
correctness of a data structure distributed on the network. The verification of
this certification must be local: a node typically sees only its neighbors in
the network. The main measure of performance of a certification is the size of
its certificates.
  In 2011, G\"o\"os and Suomela identified $\Theta(\log n)$ as a special
certificate size: below this threshold little is possible, and several key
properties do have certifications of this type. A certification with such small
certificates is now called a \emph{compact local certification}, and it has
become the gold standard of the area, similarly to polynomial time for
centralized computing. A major question is then to understand which properties
have $O(\log n)$ certificates, or in other words: what is the power of compact
local certification?
  Recently, a series of papers have proved that several well-known network
properties have compact local certifications: planarity, bounded-genus, etc.
But one would like to have more general results, \emph{i.e.} meta-theorems. In
the analogue setting of polynomial-time centralized algorithms, a very fruitful
approach has been to prove that restricted types of problems can be solved in
polynomial time in graphs with restricted structures. These problems are
typically those that can be expressed in some logic, and the graph structures
are whose with bounded width or depth parameters. We take a similar approach
and prove the first meta-theorems for local certification. (See the abstract of
the pdf for more details.)
