Triangulating a Surface with a Prescribed Graph

We prove that the following problem is NP-complete: Given a graph G, does there exist a surface S such that G triangulates S? In particular, the graph genus problem is NP-complete even when we consider only embeddings of the smallest possible genus allowed by Euler′s formula. We also prove that it is NP-complete to decide if a graph contains a collection of distinct triangles covering each edge twice a property which is necessary for the graph to triangulate a surface). The triangulation problem is reduced to the following problem, which we prove is NP-complete: Given a cubic bipartite graph G, does G contain two Hamiltonian cycles with a perfect matching in common?