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.

中文翻译：

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图分解的本地认证和对未成年人自由类的应用

本地认证包括为网络节点分配标签以证明满足某些给定属性，从而可以在本地检查标签。在过去的几年中，图类的认证受到了相当大的关注。目标是证明图 $G$ 属于给定的图类~$\mathcal{G}$。这种带有 $O(\log n)$（其中 $n$ 是网络的大小）标签的证明存在于树、平面图和嵌入在表面上的图。Feuilloley 等人。询问这是否可以扩展到由一组有限的禁止未成年人定义的任何类别的图。在这项工作中，我们开发了用于图证明的新分解工具，并应用它们来证明对于每一个足够小的小 $H$，$H$-无小图确实可以用大小为 $O(\log n )$。