Local certification of graphs on surfaces
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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 certificate 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.
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