Local Certification of Graph Decompositions and Applications to Minor-Free Classes

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