Local certification of graphs on surfaces

TL;DR

This paper presents a concise proof for local certification schemes for graphs on any surface, encoding rotation systems and spanning trees to verify embeddability efficiently.

Contribution

It provides a simplified proof for local certification of surface-embeddable graphs, extending previous results from planar graphs to all surfaces.

Findings

Short proof for local certification schemes on surfaces

Encoding rotation systems and spanning trees locally

Efficient verification of graph genus

Abstract

A proof labelling scheme for a graph class $\mathcal{C}$ is an assignment of certificates to the vertices of any graph in the class $\mathcal{C}$, such that upon reading its certificate and the certificates of its neighbors, every vertex from a graph $G\in \mathcal{C}$ accepts the instance, while if $G\not\in \mathcal{C}$, for every possible assignment of certificates, at least one vertex rejects the instance. It was proved recently that for any fixed surface $\Sigma$, the class of graphs embeddable in $\Sigma$ has a proof labelling scheme in which each vertex of an $n$-vertex graph receives a certificate of at most $O(\log n)$ bits. The proof is quite long and intricate and heavily relies on an earlier result for planar graphs. Here we give a very short proof for any surface. The main idea is to encode a rotation system locally, together with a spanning tree supporting the local computation of the genus via Euler's formula.