# Local certification of graphs on surfaces

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

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