A proof-labeling scheme (PLS) for a boolean predicate $\Pi$ on labeled graphs is a mechanism used for certifying the legality with respect to $\Pi$ of global network states in a distributed manner. In a PLS, a certificate is assigned to each processing node of the network, and the nodes are in charge of checking that the collection of certificates forms a global proof that the system is in a correct state, by exchanging the certificates once, between neighbors only. The main measure of complexity is the size of the certificates. Many PLSs have been designed for certifying specific predicates, including cycle-freeness, minimum-weight spanning tree, planarity, etc. In 2021, a breakthrough has been obtained, as a meta-theorem stating that a large set of properties have compact PLSs in a large class of networks. Namely, for every $\mathrm{MSO}_2$ property $\Pi$ on labeled graphs, there exists a PLS for $\Pi$ with $O(\log n)$-bit certificates for all graphs of bounded tree-depth. This result has been extended to the larger class of graphs with bounded {tree-width}, using certificates on $O(\log^2 n)$ bits. We extend this result even further, to the larger class of graphs with bounded clique-width, which, as opposed to the other two aforementioned classes, includes dense graphs. We show that, for every $\mathrm{MSO}_1$ property $\Pi$ on labeled graphs, there exists a PLS for $\Pi$ with $O(\log^2 n)$ bit certificates for all graphs of bounded clique-width.

翻译：针对标记图上布尔谓词$\Pi$的证明标签方案是一种用于以分布式方式认证全局网络状态关于$\Pi$合法性的机制。在PLS中，每个网络处理节点被分配一个证书，节点通过仅在邻居间交换一次证书，来检查这些证书集合是否构成系统处于正确状态的全局证明。主要复杂度衡量指标是证书大小。人们已设计出许多用于认证特定谓词（如无环性、最小权重生成树、可平面性等）的PLS。2021年取得突破性进展，得到一个元定理：对于一大类网络中的大量属性，存在紧凑的PLS。具体而言，对于标记图上的每个$\mathrm{MSO}_2$属性$\Pi$，存在一个针对该$\Pi$的PLS，其对于所有有界树深度的图使用$O(\log n)$比特的证书。该结果已被推广至更大的有界树宽图类，但需使用$O(\log^2 n)$比特的证书。我们进一步将该结果推广至更大的有界团宽图类——与前两者不同，此类图包含稠密图。我们证明：对于标记图上的每个$\mathrm{MSO}_1$属性$\Pi$，存在一个针对该$\Pi$的PLS，其对于所有有界团宽的图使用$O(\log^2 n)$比特的证书。