  Local certification consists in assigning labels to the nodes of a network to
certify that some given property is satisfied, in such a way that the labels
can be checked locally. In the last few years, certification of graph classes
received a considerable attention. The goal is to certify that a graph $G$
belongs to a given graph class~$\mathcal{G}$. Such certifications with labels
of size $O(\log n)$ (where $n$ is the size of the network) exist for trees,
planar graphs and graphs embedded on surfaces. Feuilloley et al. ask if this
can be extended to any class of graphs defined by a finite set of forbidden
minors. In this work, we develop new decomposition tools for graph
certification, and apply them to show that for every small enough minor $H$,
$H$-minor-free graphs can indeed be certified with labels of size $O(\log n)$.
We also show matching lower bounds with a simple new proof technique.
