Distance-based certification for leader election in meshed graphs and local recognition of their subclasses

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.