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

Abstract

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