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Local certification consists in assigning labels to the nodes of a network tocertify that some given property is satisfied, in such a way that the labelscan be checked locally. In the last few years, certification of graph classesreceived a considerable attention. The goal is to certify that a graph $G$belongs to a given graph class~$\mathcal{G}$. Such certifications with labelsof 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 thiscan be extended to any class of graphs defined by a finite set of forbiddenminors. In this work, we develop new decomposition tools for graphcertification, 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.

Local certification of graph decompositions and applications to minor-free classes