Self-dual embeddings of composition graphs

A triangulation of an embedded graph G is the graph obtained from G by adding a vertex in each region and joining it to each vertex in the boundary of the region. If a graph G has a self-dual embedding in a surface S, we may use current-voltage duality theory on the triangulation of G to construct a self-dual embedding of the composition graph G(n) = G[K̄_n]. This method will be applied to construct self-dual embeddings of some families of graphs such as the composition of many bipartite graphs including the Cartesian product of even cycles.