A Locally Checkable Labeling (LCL) is a specification describing a set of labels that are valid with respect to a set of conditions that characterize a local part of a solution to a global problem. Conditions can only refer to nodes and labels within a constant radius neighborhood of each node. This work studies local labeling schemes whose global consistency implies solutions to two classical problems: leader election and spanning tree construction. For each problem, we present a local labeling scheme using one bit per edge or equivalently $O(Δ)$ bits per node (where $Δ$ is the maximum degree in the graph), with conditions checkable within the graph induced by the one neighborhood of each node. For leader election, we show that global satisfaction of the conditions implies the existence of a unique sink in the graph, which we define to be a leader, while in the spanning tree setting it implies that a specific subset of edges induces a spanning tree rooted at a given node. We show those implications for $K_4$-free dismantlable and chordal graphs in the former case and for dismantlable graphs in the latter, assuming a root is given. For chordal graphs, the labeling implying a unique sink additionally induces an acyclic orientation. This property is not generally locally verifiable with constant-size labels in arbitrary graphs. To the best of our knowledge, these are the first local labeling results tailored to ($K_4$-free) dismantlable graphs, potentially highlighting structural properties useful for designing LCLs for additional problems. Finally, we present a generic transformation that converts any local labeling scheme 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）是一种描述标记集合的规范，这些标记相对于刻画全局问题解决方案局部部分的条件集合是有效的。条件仅能引用每个节点常数半径邻域内的节点和标记。本研究探讨了其全局一致性蕴含两个经典问题——领导者选举和生成树构造——解决方案的局部标记方案。针对每个问题，我们提出了一种每边使用一比特（等价于每节点$O(Δ)$比特，其中$Δ$为图中最大度数）的局部标记方案，其条件可在每个节点的一跳邻域诱导子图中验证。对于领导者选举，我们证明条件的全局满足性意味着图中存在唯一汇点（我们将其定义为领导者）；而在生成树场景中，则意味着特定边子集能诱导出以给定节点为根的生成树。我们分别在无$K_4$可拆解图与弦图（前者）以及可拆解图（后者）中证明了这些蕴含关系，并假设根节点已给定。对于弦图，蕴含唯一汇点的标记方案还会诱导一个无环定向。该性质在任意图中通常无法用常数大小标记进行局部验证。据我们所知，这是首个针对（无$K_4$）可拆解图定制的局部标记研究成果，可能揭示了有助于为其他问题设计LCL的结构特性。最后，我们提出一种通用转换方法，在假设Gouda公平调度器的前提下，仅通过增加一个额外状态即可将任意局部标记方案转换为静默自稳定算法。该转换可能具有独立的研究价值。