The genus of a graph is a fundamental parameter of a graph for which
reconstructibility is not yet proved. In this talk, I consider cellular
embeddings of a one-vertex graph into an orientable surface. I will show that
the genus of such a graph G can be reconstructed from the set of graphs G-e,
where e is an edge of G, with their corresponding cellular embeddings. The
main point of the proof is a reduction to the vertex-reconstructibility of
the determinant of the adjacency matrix of a graph. The later was proved by
W.Tutte in 1976. An approach to the general problem of
edge-reconstructibility of genus will be discussed.