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

TL;DR

This paper introduces local proof labeling schemes for leader election and class recognition in meshed graphs, leveraging distance verification and topological properties to achieve efficient, local algorithms for complex graph classes.

Contribution

It presents novel 2- and 3-local proof labeling schemes for leader election and subclass recognition in meshed graphs, utilizing distance and topological verifications.

Findings

Efficient local verification of distances in meshed graphs.

Recognition schemes for various subclasses using local properties.

Constant-size labels for leader election based on modular distance labels.

Abstract

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.