In this paper, we present a 2-local proof labeling scheme with labels in $\{ 0,1,2\}$ for leader election in anonymous meshed graphs. Meshed graphs form a general class of graphs defined by a distance condition. They comprise several important classes of graphs, which have long been the subject of intensive studies in metric graph theory, geometric group theory, and discrete mathematics: median graphs, bridged graphs, chordal graphs, Helly graphs, dual polar graphs, modular, weakly modular graphs, and basis graphs of matroids. We also provide 3-local proof labeling schemes to recognize these subclasses of meshed graphs using labels of size $O(\log D)$ (where $D$ is the diameter of the graph). To establish these results, we show that in meshed graphs, we can verify locally that every vertex $v$ is labeled by its distance $d(s,v)$ to an arbitrary root $s$. To design proof labeling schemes to recognize the subclasses of meshed graphs mentioned above, we use this distance verification to ensure that the triangle-square complex of the graph is simply connected and we then rely on existing local-to-global characterizations for the different classes we consider. To get a proof-labeling scheme for leader election with labels of constant size, we then show that we can check locally if every $v$ is labeled by $d(s,v) \pmod{3}$ for some root $s$ that we designate as the leader.

翻译：本文提出了一种用于匿名网状图中领导者选举的2-局部证明标记方案，其标记集为$\{ 0,1,2\}$。网状图是由距离条件定义的一类广义图结构。它包含多个在图度量理论、几何群论和离散数学中长期被深入研究的的重要图类：中位图、桥接图、弦图、Helly图、对极图、模图、弱模图以及拟阵的基图。我们还提供了3-局部证明标记方案，通过大小为$O(\log D)$的标记（其中$D$为图的直径）来识别这些网状图的子类。为证明这些结果，我们展示了在网状图中可以局部验证每个顶点$v$是否被标记为其到任意根节点$s$的距离$d(s,v)$。为设计识别上述网状图子类的证明标记方案，我们利用该距离验证来确保图的三角-方形复形是单连通的，并依赖现有针对所考虑各类图的局部-全局特征刻画。为获得具有恒定大小标记的领导者选举证明标记方案，我们进一步证明可以局部检验每个$v$是否被标记为$d(s,v) \pmod{3}$，其中根节点$s$被指定为领导者。