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

TL;DR

This work develops distance- and modulo-3 based proof labeling schemes for leader election and graph recognition in meshed graphs and their subclasses. It introduces a 2-local distance certification for all meshed graphs, with sharper 1-local certificates for bridged and Helly graphs, and a 2-local distance-mod-3 certificate that yields constant-size leader election labels; the approach uses a distance-based verification to connect to simply connected triangle-square complexes. The authors then leverage Cartan-Hadamard-type local-to-global characterizations to obtain $O(\log D)$-sized local recognitions for many meshed subclasses and 1–3-local recognition protocols across a wide array of graph families, including basis graphs of matroids. Together, these results unify and extend prior certificate constructions (e.g., for chordal/grid graphs) and provide practical, scalable local certificates for leader election and class recognition in large graph families, while outlining natural limitations and directions for future work.

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.

Figures (1)

• Figure 1: The triangle, quadrangle and weak quadrangle conditions

Theorems & Definitions (42)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 5
• Theorem 6
• Definition 7: Meshed and weakly modular graphs
• Lemma 8: Ch_metric
• Theorem 9
• proof