We study verification (decision) problems for graph properties in distributed networks under the locally checkable labeling framework, where nodes use labels (proofs) and local neighborhoods to decide acceptance or rejection. Our focus is twofold. First, we study cycle detection. While it is known that this can be verified using 3 labels with access to the 1-hop neighborhood, we introduce a novel gadget that encodes direction along a path using only 2 labels and access to a 3-hop neighborhood. This yields a cycle-detection labeling scheme with just 2 labels and may be of independent interest. Second, we consider adversarially corrupted labelings, where each node has access to a local neighborhood within which a fraction of nodes may receive erroneous labels. We introduce a general algorithmic framework, called refix, that transforms a base verification algorithm for a property P operating on labels within a d-hop neighborhood into one that tolerates up to i erroneous labels within a radius d+2i, by accessing a d+2i-hop neighborhood. We demonstrate applications to cycle detection, cycle absence, and bipartiteness, and provide lower bounds relating the number of errors to the required neighborhood size.

翻译：我们研究了分布式网络中图性质的验证（判定）问题，其框架基于局部可检验标记（locally checkable labeling），其中节点利用标记（证明）和局部邻域来决定接受或拒绝。我们的研究重点分为两个方面。首先，我们研究了环检测问题。虽然已知可以通过使用3个标记并访问1跳邻域来实现验证，但我们引入了一种新颖的构造机制，该机制仅使用2个标记并访问3跳邻域即可沿路径编码方向。这产生了一种仅需2个标记的环检测标记方案，且该方案可能具有独立的研究价值。其次，我们考虑了对抗性损坏标记的情形，其中每个节点可访问的局部邻域内可能存在一定比例的节点收到错误标记。我们引入了一种通用算法框架，称为refix，该框架能够将针对性质P、基于d跳邻域内标记运行的基础验证算法，转化为一种通过访问d+2i跳邻域、可容忍半径为d+2i范围内最多i个错误标记的算法。我们展示了该框架在环检测、无环性判定以及二部性判定中的应用，并给出了关于错误数量与所需邻域大小之间关系的下界。