A Locally Checkable Labeling (LCL) is a distributed constraint satisfaction problem defined on a bounded-degree graph that relates a finite set of input labels to a finite set of output labels through a finite set of locally checkable constraints. In this work we define labels and local constraints that encode solutions to two classical problems: leader election and spanning tree construction. It is known that leader election cannot be expressed as an LCL in arbitrary graphs using constant-size labels. In fact, it is known that there does not exist a finite set of labels and local constraints for leader election even for the class of rings. On the other hand, there exists a finite set of labels and local constraints characterizing leader election on trees. In this work, we prove that there exists a finite set of labels and local constraints for leader election also in the much larger class of dismantlable graphs. Our labels need one bit per edge or equivalently $O(Δ)$ bits per node (where $Δ$ is the maximum degree in the graph) and are checkable within the graph induced by the 1-neighborhood of each node. To the best of our knowledge, these are the first local labeling results tailored to dismantlable graphs, potentially highlighting structural properties useful for designing labels and constraints for additional LCL problems. Finally, we present a generic transformation that converts any finite set of labels and local constraints into a silent self-stabilizing algorithm by adding only one extra state, assuming a Gouda fair scheduler. This transformation may be of independent interest.

翻译：局部可检查标记（LCL）是在有界度图上定义的一种分布式约束满足问题，通过有限个局部可检查约束将有限个输入标签关联到有限个输出标签。本文定义了编码两种经典问题（领导者选举和生成树构建）解的标签与局部约束。已知在任意图中使用常数大小标签无法将领导者选举表达为LCL问题。事实上，即使在环类图中，也不存在针对领导者选举的有限标签集和局部约束。另一方面，树上存在刻画领导者选举的有限标签集与局部约束。本文证明，在更大的可解体图类中，也存在用于领导者选举的有限标签集与局部约束。我们的标签每条边仅需1比特（等价于每节点$O(Δ)$比特，其中$Δ$为图的最大度数），且可在每个节点1邻域诱导的子图内完成检查。据我们所知，这是首项专门针对可解体图的局部标记结果，可能揭示有助于为其他LCL问题设计标签与约束的结构性质。最后，我们提出一种通用转换方法：在假设Gouda公平调度器的情况下，仅需增加一个额外状态，即可将任何有限标签集与局部约束转化为静默自稳定算法。该转换方法可能具有独立的研究价值。