Local certification of graphs on surfaces

Abstract

A proof labelling scheme for a graph class C is an assignment of certificates to the vertices of any graph in the class C, such that upon reading its certificate and the certificates of its neighbors, every vertex from a graph G∈C accepts the instance, while if G∉C, for every possible assignment of certificates, at least one vertex rejects the instance. It was proved recently that for any fixed surface Σ, the class of graphs embeddable in Σ 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.