An optimal construction for complete graph embeddings with duals of low connectivity

We describe a construction for embeddings of complete graphs where the dual has a cutvertex and the genus of the embedding is close to the minimum genus of the primal graph. When the number of vertices is congruent to 2 or 5 modulo 12, we further guarantee that the dual is simple and that the genus of the resulting embeddings matches a lower bound of Brinkmann, Noguchi, and Van den Camp, showing that their lower bound is tight infinitely often.