Detecting specific structures in a network has been a very active theme of research in distributed computing for at least a decade. In this paper, we start the study of subgraph detection from the perspective of local certification. Remember that a local certification is a distributed mechanism enabling the nodes of a network to check the correctness of the current configuration, thanks to small pieces of information called certificates. Our main question is: For a given graph $H$, what is the minimum certificate size that allows checking that the network does not contain $H$ as a (possibly induced) subgraph? We show a variety of lower and upper bounds, uncovering an interesting interplay between the optimal certificate size, the size of the forbidden subgraph, and the locality of the verification. Along the way we introduce several new technical tools, in particular what we call the \emph{layered map}, which is not specific to forbidden subgraphs and that we expect to be useful for certifying many other properties.

翻译：检测网络中的特定结构至少十年来一直是分布式计算研究中非常活跃的主题。本文从局部认证的角度出发，开启了对子图检测的研究。局部认证是一种分布式机制，使得网络中的节点能够借助称为证书的小信息片段来检查当前配置的正确性。我们的主要问题是：对于给定图 $H$，允许检查网络不包含 $H$ 作为（可能诱导的）子图的最小证书大小是多少？我们展示了一系列下界和上界，揭示了最优证书大小、禁止子图大小以及验证局部性之间的有趣相互作用。在此过程中，我们引入了若干新的技术工具，特别是我们称之为“分层映射”的工具，它并非特定于禁止子图，我们希望它对于证明许多其他性质也能发挥有效作用。