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.

翻译：本地认证包括向网络的节点分配标签,以证明某些特定属性得到满足, 从而可以在当地检查标签。 在过去几年里, 图表类的认证受到相当重视。 目标是证明图表$G$属于特定图形类 ~ $\ mathcal{G}$ 。 具有大小为 $O( log n) 的认证( 美元是网络大小的 ) 。 嵌入表面的树木、 平面图和图表 。 Feuilloley 等人 询问, 是否可以将此扩展至由一组受禁未成年人定义的任何类型的图表 。 在这项工作中, 我们开发了用于图形认证的新的分解工具, 并应用这些工具来显示, 对于每个小小小的, $H$( $- h$- minor- fin) 图形, 确实可以用大小为 $O( log n) 的标签进行认证 。 我们还展示了与简单的新验证技术匹配的下框 。