Local Certification of Vertex and Edge Connectivity

2026-07-15

AI Summary

This work addresses the local certification of $k$-vertex-connectivity and $k$-edge-connectivity in graphs, where short certificates are assigned to vertices so that each node can verify global connectivity using only local neighborhood information. The authors present the first general-purpose local certification scheme applicable to arbitrary $k$, overcoming prior limitations restricted to small values of $k$. By integrating combinatorial tools such as branch decompositions, Eulerian subgraphs, and independent spanning trees, they devise efficient protocols grounded in structural graph properties. For $k \geq 3$, they establish a tight $O_k(\log n)$-bit upper bound for edge-connectivity certification in general graphs, matching the known lower bound; constant-size certificates are achieved for sparse graph classes. Additionally, they prove that certifying 2-vertex-connectivity in general graphs requires certificates of size $\Omega(\log \log^* n)$.

Abstract

Local certification is a framework for verifying global graph properties using only local information. In this model, a prover assigns short labels, called certificates, to the vertices of a graph. Each vertex then exchanges certificates with its neighbors and performs a purely local check to determine whether the graph satisfies the desired property. This line of research has led to efficient certification schemes for a broad range of graph classes, including minor-closed families, topological graph classes, and graphs defined by forbidden subgraphs. In this paper, we study the local certification of graph connectivity. Prior work by Bousquet, Feuilloley, and Pierron (JPDC 2024) showed that $2$-vertex-connectivity, $2$-edge-connectivity, and $3$-vertex-connectivity admit $O(\log n)$-bit certificates, leveraging structural characterizations such as ear decompositions. We go substantially beyond these cases and investigate general $k$-vertex-connectivity and $k$-edge-connectivity. We develop new approaches that exploit connections between connectivity and combinatorial structures, including branchings, Eulerian subgraphs, and independent spanning trees. For $k$-edge-connectivity, we obtain an $O_k(\log n)$-bit certification scheme and prove a matching $Ω_k(\log n)$ lower bound for every $k\ge 3$. The lower bound also applies to $k$-vertex-connectivity. For $k$-vertex-connectivity, we obtain $\tilde{O}_k(\sqrt{n})$-bit certificates for every $k$ under a conjecture of Itai and Zehavi. We further show that, for $k=2$, the logarithmic barrier can be broken on sparse graph classes: $2$-edge-connectivity admits constant-size certificates in bounded-expansion graphs, and $2$-vertex-connectivity admits constant-size certificates in bounded-degree graphs. In contrast, for $2$-vertex-connectivity in general graphs, we prove an $Ω(\log(\log^\ast n))$-bit lower bound.

Problem

• local certification
• k-vertex-connectivity
• k-edge-connectivity
• graph connectivity
• distributed verification

Innovation

• local certification
• k-connectivity
• combinatorial structures
• certificate complexity
• sparse graph classes